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arxiv: 2405.15466 · v2 · pith:TMEDRAOVnew · submitted 2024-05-24 · 🧮 math.RT

Skew-group A_(infty)-categories as Fukaya categories of orbifolds

Claire Amiot , Pierre-Guy Plamondon This is my paper

Pith reviewed 2026-05-24 01:30 UTC · model grok-4.3

classification 🧮 math.RT
keywords skew-group A-infinity categoriespartially wrapped Fukaya categoriesorbifold surfacesgraded curves with taggingstilting objectsskew-gentle algebrasderived equivalences
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The pith

Skew-group A_∞-categories serve as the partially wrapped Fukaya categories of orbifold surfaces from order-two actions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines skew-group A_∞-categories by direct analogy with skew-group algebras and positions them as the algebraic stand-in for the partially wrapped Fukaya category of an orbifold surface. Indecomposable objects are classified by graded curves that carry signs or taggings at the orbifold points. Morphisms are computed for a distinguished class of these objects. The classification is then applied to identify tilting objects and to produce algebras derived-equivalent to skew-gentle algebras.

Core claim

We define the notion of a skew-group A_∞-category and let it play the role of the partially wrapped Fukaya category of an orbifold surface. We classify indecomposable objects in terms of graded curves with signs, or taggings, at orbifold points. We compute morphisms between a class of objects, and we use this to describe tilting objects and find algebras derived-equivalent to skew-gentle algebras.

What carries the argument

Skew-group A_∞-category, constructed by analogy with skew-group algebras to encode an order-two group action on the surface.

If this is right

  • Indecomposable objects correspond to graded curves equipped with taggings at orbifold points.
  • Morphisms between a class of objects admit explicit algebraic descriptions.
  • Tilting objects in the category can be described combinatorially via the tagged curves.
  • Algebras derived-equivalent to skew-gentle algebras arise directly from the tilting objects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The tagged-curve classification supplies a combinatorial model that can be used to compute categorical invariants without returning to the geometric surface.
  • The derived-equivalence results indicate that skew-gentle algebras can be realized geometrically as endomorphism algebras in orbifold Fukaya categories.

Load-bearing premise

The skew-group A_∞-category defined by the authors correctly models the partially wrapped Fukaya category of the orbifold surface obtained from the order-two action.

What would settle it

An explicit geometric computation of morphism spaces or indecomposable objects in the partially wrapped Fukaya category of a concrete orbifold surface that fails to match the algebraic description given by the skew-group category.

Figures

Figures reproduced from arXiv: 2405.15466 by Claire Amiot, Pierre-Guy Plamondon.

Figure 2
Figure 2. Figure 2: A full system of arcs on a sphere with three boundary components Definition 3.9. An M-segment a on (S, M, η) is an embedding of [0, 1] → M oriented with the reverse orientation of the boundary of S. Given two M-segments a and b such that a(1) = b(0), we define ba = a.b to be the unique M-segment such that a.b(0) = a(0) and a.b(1) = b(1) [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: Intersection of G-invariant graded arcs Proposition 4.6. Let X and Y be G-invariant graded arcs intersecting in a fixed point as in [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 12
Figure 12. Figure 12: c ga gc a gb b Z gZ × × × P Q Y X X (γ, f) p q [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 16
Figure 16. Figure 16: Index between two graded arcs intersecting at an orb￾ifold point on the orbifold surface S/G • if γ is not connected to an orbifold point, then its preimage on S is a union of two (graded) arcs in the same G-orbit. These are associated to objects X and gX in Tw A. We define X(γ) := {X} ⊕ {gX}. • if γ has one endpoint on the boundary and another on an orbifold point with tag ±, then its lift to S is a G-in… view at source ↗
Figure 17
Figure 17. Figure 17: Example of a dissection. The circle arcs around orb￾ifold points illustrate the bijection with pieces of type (3). into a marked point, and filling each boundary component with a disc, keeping the boundary edges between marked points (unmarked boundary components thus dis￾appear). If χ is the Euler characteristic of the surface, then χ = 2 − 2g¯. Moreover, if the surface is cut into discs, then χ is the n… view at source ↗
Figure 18
Figure 18. Figure 18: Foliations on polygons of types (1) and (2) Case 2: triangles with at least one orbifold point in their boundary. Those triangles are lifts of pieces of type (3) in Definition 5.12. Such a triangle is associated to one of the orbifold point in its boundary by the bijection in the definition of a dissection. If we denote by × this orbifold point, then the foliation in this triangle is given as follows: × … view at source ↗
Figure 19
Figure 19. Figure 19: Foliation on polygon of type (3) All these foliations glue together and thus define a line field η on S. Moreover, by contruction and up to homotopy, we can assume that η is G-invariant. Then the map X of Definition 5.9 sends ∆ to an object which can be seen to be tilting using Theorem 5.10. □ We now define an algebra from a tagged dissection. Definition 5.15. Let S/G be an orbifold surface with a tagged … view at source ↗
read the original abstract

We study the partially wrapped Fukaya category of a surface with boundary with an action of a group of order two. Inspired by skew-group algebras and categories, we define the notion of a skew-group $A_\infty$-category and let it play the role of the partially wrapped Fukaya category of an orbifold surface. We classify indecomposable objects in terms of graded curves with signs, or taggings, at orbifold points. We compute morphisms between a class of objects, and we use this to describe tilting objects and find algebras derived-equivalent to skew-gentle algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines the notion of a skew-group A_∞-category, modeled on skew-group algebras, and proposes it as a model for the partially wrapped Fukaya category of an orbifold surface obtained from a Z/2-action on a surface with boundary. It classifies indecomposable objects as graded curves equipped with taggings (signs) at orbifold points, computes morphism spaces between certain objects, describes tilting objects, and identifies algebras derived-equivalent to skew-gentle algebras.

Significance. If the proposed identification holds, the work supplies an algebraic framework for studying Fukaya categories of orbifolds, yielding explicit classifications and derived equivalences that connect symplectic geometry to the representation theory of skew-gentle algebras. The purely algebraic classification and tilting results stand independently and may be of interest even without the geometric interpretation.

major comments (2)
  1. [Introduction] Introduction (paragraph 2) and the definition of skew-group A_∞-category: the manuscript asserts that this algebraically defined category 'plays the role of' the partially wrapped Fukaya category of the orbifold without a comparison theorem, invariance statement under the group action, or explicit verification that the A_∞ operations coincide with counts of holomorphic curves in the orbifold. This identification is load-bearing for the geometric interpretation claimed in the title and abstract.
  2. [Section on classification of indecomposables] The classification of indecomposables (graded curves with taggings) and the subsequent computation of morphisms are presented as consequences of the new definition; however, without a proof that the definition satisfies the A_∞ relations or matches geometric data, it is unclear whether these algebraic objects correspond to actual objects in the Fukaya category.
minor comments (2)
  1. [Introduction] Notation for the group action and the resulting orbifold points should be introduced with a short diagram or explicit local model to aid readers unfamiliar with orbifold surfaces.
  2. The paper would benefit from a brief comparison table or statement clarifying how the skew-group A_∞-category differs from or extends existing constructions such as equivariant Fukaya categories or orbifold Fukaya categories in the literature.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and for highlighting the distinction between the algebraic construction and its geometric motivation. The manuscript develops the skew-group A_∞-category as an algebraic object modeled on skew-group algebras, with the orbifold Fukaya interpretation serving as motivation rather than a proven equivalence. We address the two major comments below and indicate the revisions we are prepared to make.

read point-by-point responses

  1. Referee: [Introduction] Introduction (paragraph 2) and the definition of skew-group A_∞-category: the manuscript asserts that this algebraically defined category 'plays the role of' the partially wrapped Fukaya category of the orbifold without a comparison theorem, invariance statement under the group action, or explicit verification that the A_∞ operations coincide with counts of holomorphic curves in the orbifold. This identification is load-bearing for the geometric interpretation claimed in the title and abstract.

    Authors: We agree that the paper does not contain a comparison theorem or explicit verification that the A_∞ operations match holomorphic curve counts. The skew-group A_∞-category is defined purely algebraically by extending the skew-group construction to A_∞-categories, and the claim that it 'plays the role of' the Fukaya category of the orbifold is presented as a modeling proposal inspired by the skew-group algebra case and the expected structure of orbifold Fukaya categories. No invariance under the group action or geometric matching is proven. We will revise the introduction, abstract, and title to state explicitly that the identification is conjectural/motivational and that the results are algebraic. The referee's observation that the algebraic classification and tilting results stand independently is correct. revision: partial

  2. Referee: [Section on classification of indecomposables] The classification of indecomposables (graded curves with taggings) and the subsequent computation of morphisms are presented as consequences of the new definition; however, without a proof that the definition satisfies the A_∞ relations or matches geometric data, it is unclear whether these algebraic objects correspond to actual objects in the Fukaya category.

    Authors: The classification of indecomposables as graded curves with taggings and the morphism computations are derived strictly from the algebraic definition; we verify the A_∞ relations hold for the defined operations in the section introducing the skew-group A_∞-category. These results are therefore valid as statements about the algebraic category. We acknowledge, however, that the paper provides no proof that the definition matches geometric data from the orbifold Fukaya category, so the correspondence to actual Fukaya objects remains unverified. We will add a clarifying paragraph in the classification section and a short discussion subsection noting the scope of the algebraic results. revision: partial

standing simulated objections not resolved
  • Providing a full comparison theorem or invariance statement relating the algebraically defined skew-group A_∞-category to the partially wrapped Fukaya category of the orbifold surface, which lies outside the algebraic focus of the present manuscript and would require substantial additional geometric work.

Circularity Check

0 steps flagged

New definition by analogy; algebraic consequences self-contained

full rationale

The paper introduces the skew-group A_∞-category as a new definition inspired by skew-group algebras, then assigns it the modeling role for the orbifold Fukaya category. All subsequent results (classification of indecomposables via graded curves with taggings, morphisms, tilting objects, derived equivalences to skew-gentle algebras) are direct algebraic consequences of this definition. No equations reduce a claimed prediction back to a fitted input, no uniqueness theorem is imported from self-citation, and no ansatz is smuggled via prior work by the same authors. The identification with the geometric category is an explicit modeling assumption rather than a derived equality that loops on itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Ledger constructed from abstract only; full paper may contain additional technical assumptions.

axioms (1)
  • standard math Standard axioms and properties of A_∞-categories and partially wrapped Fukaya categories hold.
    The construction relies on the established theory of these structures.
invented entities (1)
  • skew-group A_∞-category no independent evidence
    purpose: To serve as the partially wrapped Fukaya category of an orbifold surface
    Newly introduced definition inspired by skew-group algebras.

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Reference graph

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