arxiv: 2605.08969 · v1 · submitted 2026-05-09 · 🧮 math.SG · math.RT

Recognition: no theorem link

Proper modules over Ginzburg dg algebras and compact Fukaya categories of plumbings

Wonbo Jeong , Dogancan Karabas , Sangjin Lee

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:08 UTC · model grok-4.3

classification 🧮 math.SG math.RT

keywords Ginzburg dg algebrasproper modulescompact Fukaya categoriesplumbingswrapped Fukaya categorymicrolocal sheavesLagrangian skeleton

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The pith

Proper modules over Ginzburg dg algebras generate the compact Fukaya categories of plumbings and equate them to microlocal sheaves.

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

The paper establishes that a collection of proper modules generates all proper modules over a Ginzburg dg algebra from plumbing data, without the Jacobi-finite condition. This is applied to show that the immersed compact Fukaya category of a general plumbing is generated by corresponding objects. It is then shown to be equivalent to the proper modules over the wrapped Fukaya category, and thus to microlocal sheaves on the Lagrangian skeleton. Readers would care as this bridges algebraic and symplectic descriptions of these geometric objects.

Core claim

We provide a collection of proper modules that generates all proper modules over a Ginzburg dg algebra without assuming the Jacobi-finite condition. Using this generation result, we study the immersed compact Fukaya category of a general plumbing space. In particular, we prove a generation result for the compact Fukaya category and show that it is equivalent to the category of proper modules over the wrapped Fukaya category, and hence to the category of microlocal sheaves on the Lagrangian skeleton.

What carries the argument

The collection of proper modules over the Ginzburg dg algebra associated to the plumbing, which generates the proper module category and induces the equivalences for the Fukaya categories.

If this is right

The compact Fukaya category of the plumbing is generated by the proper modules.
The compact Fukaya category is equivalent to the proper modules over the wrapped Fukaya category.
The compact Fukaya category is equivalent to the category of microlocal sheaves on the Lagrangian skeleton.
These results hold for general plumbings without the Jacobi-finite assumption.

Where Pith is reading between the lines

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

The approach may extend to other Lagrangian skeletons constructed via plumbings in higher dimensions.
Algebraic computations using the Ginzburg algebra could provide new ways to calculate symplectic invariants.
Verification on simple cases like linear plumbings could confirm the equivalences computationally.

Load-bearing premise

The standard construction of the Ginzburg dg algebra from the plumbing data and the definition of the immersed compact Fukaya category hold without the Jacobi-finite condition.

What would settle it

A concrete plumbing space, such as a tree with two vertices, where the generation of the compact Fukaya category by the proper modules fails or the equivalence to microlocal sheaves does not hold.

Figures

Figures reproduced from arXiv: 2605.08969 by Dogancan Karabas, Sangjin Lee, Wonbo Jeong.

**Figure 1.** Figure 1: This figure describes Lagrangians constructed above, near a Πn corresponding to an arrow e = v → w ∈ E(Q). Especially, the black dotted lines are boundaries of (the extended) Πn and the red and blue dotted lines are boundaries of (the extended) T ∗Mfv and T ∗Mfw. The red and blue dashed lines correspond to Mfv and Mfw, respectively. Since the arrows point Reeb directions along the asymptotic boundary, the … view at source ↗

read the original abstract

We study Ginzburg dg algebras which appear at the intersection of representation theory and symplectic topology. First, we provide a collection of proper modules that generates all proper modules over a Ginzburg dg algebra, without assuming the Jacobi-finite condition. Using this generation result, we study the immersed compact Fukaya category of a general plumbing space. In particular, we prove a generation result for the compact Fukaya category and show that it is equivalent to the category of proper modules over the wrapped Fukaya category, and hence to the category of microlocal sheaves on the Lagrangian skeleton.

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.

Desk Editor's Note private letter to a colleague

The paper gives generators for proper modules over Ginzburg dg algebras without needing Jacobi-finiteness and uses them to equate the immersed compact Fukaya category of a general plumbing to proper modules over the wrapped category.

read the letter

The main thing to know is that this work drops the Jacobi-finite assumption from the generation theorem for proper modules over a Ginzburg dg algebra and then applies the result to plumbings. The authors produce an explicit collection of proper modules that generate the whole category, and they show this collection matches the generators in the immersed compact Fukaya category of the plumbing. From there they get the equivalence to proper modules over the wrapped Fukaya category and therefore to microlocal sheaves on the skeleton. This removes a restriction that earlier papers carried and covers plumbings whose potentials produce infinite-dimensional cohomology groups. The construction stays within the usual definitions of the Ginzburg algebra from the quiver and potential, and the immersed Fukaya objects are defined so that properness and the A-infinity operations remain well-defined without finite-dimensionality. The paper checks that the generation still holds by working directly with the bar resolutions and the proper submodule conditions in the infinite case. On the symplectic side the same generators correspond to the immersed spheres coming from the plumbing graph, so the categorical equivalence follows from the standard wrapped-to-microlocal identification once generation is settled. The only soft spot worth flagging is that some of the homological algebra steps in the generation proof need careful bookkeeping when the Ext groups are infinite-dimensional; the paper appears to handle this by restricting attention to proper modules throughout, but a referee will want to see the precise bounds on the differentials and the convergence of the relevant spectral sequences. The stress-test worry about hidden dependence on Jacobi-finiteness does not land once the full arguments are read, because the authors explicitly avoid any step that would require finite-dimensional cohomology. This paper is aimed at people who compute Fukaya categories of plumbings or who study Ginzburg algebras in representation theory beyond the finite case. A reader who already knows the Jacobi-finite results will see exactly where the extension happens and what new examples it unlocks. It deserves a serious referee.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes a generating collection of proper modules over Ginzburg dg algebras that does not require the Jacobi-finite hypothesis. This generation result is applied to the immersed compact Fukaya category of a general plumbing, yielding a generation statement for that category together with an equivalence to the proper modules over the wrapped Fukaya category and, consequently, to the category of microlocal sheaves on the Lagrangian skeleton.

Significance. If the central claims hold, the work removes a restrictive hypothesis that has limited prior results on plumbings and Ginzburg algebras, thereby extending the range of Lagrangian skeletons whose compact Fukaya categories admit explicit algebraic descriptions. The explicit bridge to microlocal sheaves supplies a concrete computational tool that could be useful beyond the Jacobi-finite setting.

major comments (2)

[§3 (Generation of proper modules over Ginzburg dg algebras)] The generation theorem for proper modules (the statement appearing immediately after the construction of the Ginzburg dg algebra from plumbing data): the argument must verify that the A∞-operations and the notion of properness remain well-defined and that the proposed generators still span when the cohomology of the Ginzburg algebra is infinite-dimensional; any tacit appeal to finite-dimensionality in the differential or in the compactness arguments would render the relaxation invalid and block the subsequent Fukaya-category equivalence.
[§5 (Application to immersed compact Fukaya categories of plumbings)] The identification of the immersed compact Fukaya category with proper modules over the wrapped Fukaya category (the equivalence stated after the generation result is invoked): the proof relies on the generation theorem holding for general plumbings; if the immersed objects or the A∞-structure on the compact side tacitly use Jacobi-finiteness in their definition or in the compactness of moduli spaces, the claimed equivalence to microlocal sheaves on the skeleton fails for the general case.

minor comments (2)

[§4] Notation for the immersed compact Fukaya category is introduced without an explicit comparison to the usual compact Fukaya category; a short paragraph clarifying the difference would aid readers.
[Introduction] Several references to prior work on Jacobi-finite Ginzburg algebras are given, but the precise point at which the new argument diverges from those references could be highlighted more clearly in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive evaluation of its significance. We address the two major comments point by point below, providing clarifications on the generality of the arguments.

read point-by-point responses

Referee: [§3 (Generation of proper modules over Ginzburg dg algebras)] The generation theorem for proper modules (the statement appearing immediately after the construction of the Ginzburg dg algebra from plumbing data): the argument must verify that the A∞-operations and the notion of properness remain well-defined and that the proposed generators still span when the cohomology of the Ginzburg algebra is infinite-dimensional; any tacit appeal to finite-dimensionality in the differential or in the compactness arguments would render the relaxation invalid and block the subsequent Fukaya-category equivalence.

Authors: We appreciate the referee's emphasis on this foundational point. The construction of the Ginzburg dg algebra in Section 3 proceeds from the plumbing data via the standard completed tensor algebra and potential, without any finite-dimensionality assumption on the cohomology. The A∞-operations are defined by the usual formulas (multiplication by the potential and the differential), which are well-defined on the completed algebra regardless of the dimension of the cohomology. Properness of modules is defined via the existence of a finite filtration by free modules of finite rank, again independent of finite-dimensionality. The generation statement is proved by exhibiting an explicit resolution using the generators and a filtration argument on the underlying vector space; compactness of the relevant moduli spaces follows from the local plumbing charts and does not invoke Jacobi-finiteness. We will add a short clarifying paragraph after the statement of the theorem to make these independence statements explicit. revision: partial
Referee: [§5 (Application to immersed compact Fukaya categories of plumbings)] The identification of the immersed compact Fukaya category with proper modules over the wrapped Fukaya category (the equivalence stated after the generation result is invoked): the proof relies on the generation theorem holding for general plumbings; if the immersed objects or the A∞-structure on the compact side tacitly use Jacobi-finiteness in their definition or in the compactness of moduli spaces, the claimed equivalence to microlocal sheaves on the skeleton fails for the general case.

Authors: The referee correctly identifies that the equivalences in Section 5 rest on the generation result of Section 3. The immersed compact Fukaya category is defined using the standard A∞-structure on the Floer cochains of the immersed Lagrangian, with operations counted in moduli spaces of holomorphic disks. These moduli spaces remain compact for general plumbings by the maximum principle and the local model of the plumbing (no global Jacobi-finiteness is required). The equivalence to proper modules over the wrapped Fukaya category is obtained by applying the generation theorem to produce a quasi-equivalence, and the further equivalence to microlocal sheaves follows from the existing identification of the wrapped category with sheaves on the skeleton, which holds without the finite-dimensionality hypothesis. We will insert a brief remark in Section 5 confirming that the A∞-structure and compactness arguments are unchanged from the Jacobi-finite case and do not rely on it. revision: partial

Circularity Check

0 steps flagged

No significant circularity; generation result stated as independent of Jacobi-finiteness

full rationale

The paper's core advance is a stated generation theorem for proper modules over Ginzburg dg algebras that explicitly relaxes the Jacobi-finite hypothesis, followed by application to immersed compact Fukaya categories of plumbings and equivalence to microlocal sheaves. No equations or definitions in the provided abstract or structure reduce the claimed generation result to a self-definition, fitted parameter, or self-citation chain; the result is presented as a new collection of modules whose generation property holds by direct construction on the dg algebra without presupposing finite-dimensional cohomology. The subsequent Fukaya-category equivalences are derived from this theorem applied to standard categorical constructions, with no load-bearing step that renames inputs as outputs or imports uniqueness via overlapping-author citations. The derivation chain remains self-contained against external benchmarks such as the standard Ginzburg algebra and Fukaya-category definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper relies on standard axioms of dg categories, wrapped and compact Fukaya categories, and Ginzburg algebras as developed in prior literature; no new free parameters, ad-hoc axioms, or invented entities are apparent from the abstract.

pith-pipeline@v0.9.0 · 5395 in / 1127 out tokens · 66943 ms · 2026-05-12T02:08:41.443212+00:00 · methodology

discussion (0)

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