Orlov and Viterbo functors in partially wrapped Fukaya categories

Zachary Sylvan

REVIEW 2 major objections 5 minor 1 cited by

Orlov and Viterbo functors in partially wrapped Fukaya categories

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a stop's swappability makes its Orlov functor spherical, and that Viterbo transfer is a homological epimorphism under stop removal.

desk verdict A genuinely new Viterbo result and a good geometric criterion for spherical Orlov functors, but the proof of the Orlov half has a shift-bookkeeping gap that a referee should check before the exact monodromy/twist identifications are trusted. read the letter →

arxiv 1908.02317 v1 pith:L7GDJAUP submitted 2019-08-06 math.SG

Zachary Sylvan This is my paper · ORCID

classification math.SG MSC 53D3753D4053D10

keywords partiallywrappedFukayacategoryOrlovfunctorViterbotransfermapsphericalswappablestophomologicalepimorphismLiouvillesectorremoval

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Partially wrapped Fukaya categories are algebraic invariants that organize the exact Lagrangians in a Liouville sector, and two natural maps between them are the Orlov functor (which sends a Lagrangian in a stop to a small disk linking the corresponding core) and the Viterbo transfer map (which morally sends a Lagrangian to its intersection with a subdomain). This paper establishes a geometric condition, the swappability of a stop, under which the Orlov functor is spherical: its twist and cotwist autoequivalences are inverse, recovering a structural pattern previously known for Landau–Ginzburg models. The paper also proves that the Viterbo transfer map is a homological epimorphism whenever the domain and subdomain satisfy stop removal, meaning that after passing to module categories it becomes a localization. These results matter because they carry two kinds of algebraic localization, sphericality and homological epimorphism, from the Weinstein setting to broader Liouville sectors, and they tie these algebraic properties to concrete contact-geometric data.

What carries the argument

The new geometric input is a swappable stop: a stop $\sigma$ in the boundary at infinity equipped with an isotopy from its positive Reeb pushoff $\sigma_+$ to its negative pushoff $\sigma_-$ that avoids $\sigma$ itself. That isotopy induces three autoequivalences: a monodromy $M_\varphi$ on the wrapped category of the fiber, a wrap-once $W_\varphi$ on the partially wrapped category of the sector, and a swap $S_\varphi$ on the sector obtained by gluing two copies of the stop. The algebraic engine is the spherical swap: an autoequivalence of a semiorthogonal gluing, an $A_\infty$-category built from two categories and a bimodule between them, that exchanges the two complementary subcategories, and Proposition 4.4 shows that a functor is spherical exactly when such a positive swap exists. The proof of Theorem 1.3 identifies $S_\varphi$ with a positive spherical swap of the Orlov functor. For the Viterbo theorem, the machinery is the sectorial construction of the transfer map together with its doubled and tripled gluings; the argument shows that the bimodule map $\Gamma^{\dagger}(V) \otimes_{W(\overline{M})} \Gamma(V) \to \Delta_{W(M_{\mathrm{in}})}$, whose being a quasi-isomorphism is equivalent to $V$ being a homological epimorphism, is induced by an isotopy of sectors, so stop removal upgrades it to a quasi-isomorphism.

What would settle it

The claim would be settled by exhibiting a stop that satisfies stop removal and admits the required positive-to-negative Reeb-pushoff isotopy but whose Orlov functor is not spherical; conversely, a non-swappable stop whose Orlov functor is spherical would show swappability is not necessary for the conclusion.

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Extended reading notes

Core claim

The central claim is that two natural functors between partially wrapped Fukaya categories have good algebraic structure under explicit geometric hypotheses. On the Orlov side, Theorem 1.3 says that if $\sigma$ is a swappable stop satisfying stop removal, then the Orlov functor $\imath_\sigma: W(\overline{F}) \to W(M)$ is spherical, meaning its twist and cotwist autoequivalences are inverse, and the monodromy $M_\varphi$ and wrap-once $W_\varphi$ autoequivalences obtained from the swapping isotopy agree with the shifted dual cotwist $M'[-2]$ and the dual twist $W'$, respectively; in particular these autoequivalences are independent of the choice of isotopy. On the Viterbo side, Theorem 1.8 says that for a Liouville subdomain $M_{\mathrm{in}} \subset \overline{M}$ with both $\overline{M}$ and the completion of $M_{\mathrm{in}}$ satisfying stop removal, the Viterbo transfer $V: W(\overline{M}) \to \mathrm{Perf}\,W(M_{\mathrm{in}})$ is a homological epimorphism: its extension to module categories is a localization, and its image split-generates $W(M_{\mathrm{in}})$. Since stop removal is known for Weinstein manifolds, the theorem applies in particular when the domain and subdomain are individually Weinstein, even if the cobordism between them is not.

Load-bearing premise

The main conclusions are conditional on two existence hypotheses whose general validity is left open: the stop must admit a swappable isotopy from its positive Reeb pushoff to its negative pushoff avoiding itself, and the relevant Liouville manifolds must satisfy stop removal; if either hypothesis fails, the corresponding theorem is not claimed to hold.

Editorial extensions

If this is right

A swappable stop satisfying stop removal yields a spherical Orlov functor, so the twist and cotwist autoequivalences around the stop are inverse to one another.
The monodromy and wrap-once autoequivalences are independent of the chosen swapping isotopy; any symplectomorphism of the ambient sector that preserves the stop setwise commutes with them.
For a Liouville subdomain whose ambient domain and completion both satisfy stop removal, the Viterbo transfer map is a homological epimorphism, so its image split-generates the wrapped category of the subdomain.
When the domain and subdomain are individually Weinstein, the Viterbo transfer map is a homological epimorphism even if the cobordism between them is not Weinstein.
Passing to monodromy-invariant subdomains of a swappable stop produces new swappable stops, giving examples beyond the standard Landau–Ginzburg fiber construction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The algebraic equivalence between sphericality and existence of a positive spherical swap suggests a converse the paper does not prove: every spherical Orlov functor should arise from a swappable stop, so testing non-swappable stops for sphericality would sharpen the geometric criterion.
Theorem 1.8 recasts the open question of whether the Viterbo map is a genuine localization for individually Weinstein domain and subdomain as the compact generation of the kernel of its module-level extension, a condition one could approach through the co-cores of the cobordism.
The swap autoequivalence built from a swappable stop on a mapping torus gives a concrete symplectic mechanism for realizing any symplectomorphism-induced autoequivalence as a spherical twist, linking the paper's constructions to the broader principle that autoequivalences are spherical twists.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know about arXiv:1908.02317. The Viterbo half (Theorem 1.8) is a genuinely new result, proved by a clean comparison of doubled and tripled Viterbo sectors, and it looks structurally sound. The Orlov half (Theorem 1.3) introduces a good notion — swappable stops — but the proof's shift bookkeeping is terse enough that I want a referee to redo the gradings before I trust the exact identifications Wφ = W′ and Mφ = M′[−2]. The sphericality part is less at risk; even if a shift were off, that part would still survive.

Let me give credit where it is earned. The definition of swappable stop is the right geometric abstraction: it is what makes the Orlov functor spherical, and the paper shows that the monodromy and wrap-once autoequivalences are exactly the expected twist/cotwist expressions. The proof of Theorem 1.8 is the first written proof that Viterbo transfer is a homological epimorphism under stop removal, and the tripled-sector argument is a nice way to see the localization. The paper is honest about its assumptions: stop removal is cited for Weinstein domains, and the lack of a general criterion for swappability is acknowledged with examples.

Now the soft spots. The one that matters is in Section 4.2. In the proof of Theorem 1.3 the author compares two semiorthogonal presentations of the gluing category and says "reparametrize to remove both shifts," then later "a full rotation is a shift by 2." These are exactly the steps that determine whether the swap is A-positive and whether S² restricts to the claimed Wφ and Mφ. The stress-test note is right that an even shift error would still give a spherical swap, so the geometric content of "spherical" is safe, but the paper's sharpest claim — the equality Wφ = W′, Mφ = M′[−2] — would be off by a shift. I don't see an actual error in the lines I checked, but the computation is compressed and the conventions are easy to flip. This is a writeup gap, not a fatal one: it asks for an explicit grading check in the final version.

Other caveats are lighter. Lemma 2.2's compactness is a sketch, and Section 2.6 compares with Abouzaid–Seidel by property list rather than full proof. Both are acknowledged by the author. The paper depends on the Ganatra–Pardon–Shende sectorial framework, but that is standard now.

Bottom line: this paper deserves a serious referee. I'd send it to review, and I'd ask the author to expand the shift verification in Theorem 1.3 — a short explicit computation would settle it. Once that is done, I'd expect this to be a solid addition to the wrapped-Fukaya literature. For my own work I'd cite the Viterbo result now; I'd wait on the exact shifted equalities until the grading check is visible.

Referee Report

2 major / 5 minor

Summary. This paper studies two functors between partially wrapped Fukaya categories in the Ganatra–Pardon–Shende sectorial framework. The first is the Orlov functor associated to a stop; the paper introduces the notion of a swappable stop and proves (Theorem 1.3) that, under stop removal, a swappable stop makes the Orlov functor spherical, with the monodromy autoequivalence equal to the shifted dual cotwist and the wrap-once autoequivalence equal to the dual twist. The second is the Viterbo transfer map from a Liouville domain to a subdomain; the paper proves (Theorem 1.8) that, when the ambient domain and the completion of the subdomain satisfy stop removal, the Viterbo transfer map is a homological epimorphism, hence becomes a localization at the level of module categories and has split-generating image. The proofs are geometric: the algebraic content is organized around spherical swaps and semiorthogonal gluings, while the geometric input is isotopy invariance, forward-stopped inclusions, and the sectorial Viterbo construction.

Significance. If the results are correct, this is a substantial contribution. Theorem 1.3 gives a broad geometric criterion for sphericality of Orlov functors that goes beyond Landau–Ginzburg models, and the exact identification of monodromy and wrap-once autoequivalences with the dual cotwist and dual twist is strong and novel. Theorem 1.8 is a meaningful partial answer to Question 1.7 and appears to be the first written proof that Viterbo transfer is a homological epimorphism in the non-Weinstein cobordism setting. The paper is honest about its hypotheses, including the reliance on stop removal and on the GPS framework, and it supplies explicit geometric proofs rather than any parameter fitting or data-dependent argument. The algebraic characterization of spherical functors via spherical swaps in Proposition 4.4 is a useful standalone contribution.

major comments (2)

[§4.2, proof of Theorem 1.3] The shift bookkeeping in the proof of Theorem 1.3 is load-bearing and is not verified in the text. The proof identifies the category C from Proposition 4.4 with two directed gluings: the first parametrizes W(F) via (iΣF)∗∘ı1[1] and then 'reparametrizes to remove both shifts', while the second parametrizes W(F) via (iΣF)∗∘ı1[2]. The subsequent sentence 'a full rotation is a shift by 2' is the only justification that the two copies of W(F) correspond to the two stops with exactly the gradings needed to conclude Wφ = W′ and Mφ = M′[−2]. An even shift error would still produce a spherical swap and preserve the sphericality part of Theorem 1.3, but it would change the conclusion to Wφ ≅ W′[2k] or Mφ ≅ M′[−2+2k]. Since the exact shift in these identifications is one of the principal claims, the paper should expand this paragraph into an explicit grading computation, fixing the graded inclusions i0, i1, i2 of the An sector, the conventions of Remark 2.19, and the direction of each rotation used in Proposition 3.6 and Lemma 2.15.
[§3.4, proof of Theorem 1.8] The final step of the proof of Theorem 1.8 is compressed: the text asserts that because S◦Q and the composition (3.8)∘S◦Q are both quotients by the full subcategory W(Mbar), the map (3.8) must be a quasi-isomorphism. This is a cancellation property for localizations that is plausible but is not stated or proved, and the condensed diagram in the proof is not labeled sufficiently to check that the hypotheses of such a cancellation lemma hold. Please state the exact cancellation statement used and label the arrows and objects in the diagram, or give a direct argument that the induced map on quotients is an equivalence. As written, the last step of Theorem 1.8 is not fully transparent.

minor comments (5)

[Abstract and Introduction] The abstract says 'when the domain and subdomain are independently Weinstein', but the actual hypothesis of Theorem 1.8 is stop removal for both the ambient domain and the completion of the subdomain. Since Weinstein implies stop removal, the abstract should either state the more general hypothesis or explicitly note that it is implying the Weinstein case.
[Example 1.4] The spelling 'Landau–Ginsburg' should be 'Landau–Ginzburg'.
[Lemma 2.2] The proof of Lemma 2.2 is only a sketch and it is not always clear which parts are new and which are being quoted from [14]. A sentence saying precisely which statements from [14] are being invoked would help the reader assess the compactness claim.
[§3.4, diagram (3.9)] The diagram in (3.9) and the condensed diagram in the proof of Theorem 1.8 are visually dense and some arrows are not labeled. Adding labels such as S◦Q, (3.8), A, and i∗ directly on the arrows would make the proof substantially easier to follow.
[Remark 3.14] The statement that the autoequivalence φ in Proposition 3.13 is 'not essential' but is nevertheless needed for the comparison would benefit from a brief comment on why it does not affect the subsequent quotient argument in the proof of Theorem 1.8.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's theorems are derived from gluing, adjunction, isotopy, and algebraic spherical-swap results, not from their own conclusions.

full rationale

This is a proof paper with no fitted parameters or data. The central results—Theorem 1.3 (spherical Orlov functor and identification of monodromy/wrap-once with dual (co)twists) and Theorem 1.8 (Viterbo transfer as homological epimorphism)—are obtained by combining geometric isotopy and gluing statements with independent algebraic facts. Theorem 1.3 reduces sphericality to the existence of a spherical swap via Proposition 4.4, whose proof is algebraic and draws on Halpern-Leistner–Shipman; the new geometric input is that a swappable stop produces such a swap S_phi. The shift comparisons in the proof are explicit computations about grading conventions, not assumptions of the conclusion; even if a sign were wrong, that would be a correctness issue rather than circularity. Theorem 1.8 is proved by showing that a certain bimodule map (3.8) is a quasi-isomorphism, using stop removal and the deformation equivalence of the tripled and doubled Viterbo sectors; the definition of V via (Perf i_{sigma_1}[1])^{-1} i_{sigma_0} does not itself assert homological epimorphism. The paper relies on Ganatra–Pardon–Shende for sectorial Floer theory and gluing formulas and on the author's earlier work for partially wrapped Floer foundations; these are background results with independent proofs, not a self-citation chain that smuggles in the theorem. No equation in the paper reduces to its own input by construction, and no fitted value is renamed as a prediction. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters appear: the paper is a proof-based theory paper, and all numerical or structural choices are conventions, such as the grading shift in the Viterbo construction, rather than parameters fitted to data. The invented entities list is empty; swappable stops are a new definition, not a postulated object. The axiomatic load is carried by the Ganatra-Pardon-Shende sectorial framework, stop removal, and gluing formulas, all flagged in the text.

assumptions (5)

standard math Existence, invariance, and continuation functors for partially wrapped Fukaya categories on Liouville sectors (Ganatra-Pardon-Shende and Sylvan)
Used throughout as the framework; Sections 2.1 and 2.2 cite [14], [15], and [31] for definitions, Gromov compactness, and continuation functors rather than re-proving them.
domain assumption Stop removal holds for the manifolds in the theorems
Definition 2.16 and Example 2.17 state Weinstein manifolds satisfy stop removal; Theorem 1.8 explicitly assumes both Mbar and Min satisfy it, and Theorem 1.3 assumes the stop satisfies it.
domain assumption The gluing formulas for Liouville sectors and the semiorthogonal decomposition presentations (Proposition 3.9, Corollary 3.11)
The proof of Theorem 1.8 uses these formulas, which are proved in Section 3.3 but rely on Proposition 3.6, cited as known to experts with proof sketches.
standard math Properties of spherical functors and spherical swaps from Anno-Logvinenko and Halpern-Leistner-Shipman
Proposition 4.4 and Theorem 4.1 are cited and proved and are used to convert sphericality into existence of a positive spherical swap.
domain assumption Abouzaid-Seidel Viterbo map properties
Section 2.6 lists essential properties of the Abouzaid-Seidel transfer map from [2] instead of proving them, and uses them to identify the sectorial Viterbo map.

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Cite this review

Pith. "Pith review of Orlov and Viterbo functors in partially wrapped Fukaya categories." pith.science (2026). https://pith.science/paper/L7GDJAUP

@misc{pith2026190802317,
  author       = {Pith},
  title        = {Pith review of: Orlov and Viterbo functors in partially wrapped Fukaya categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L7GDJAUP}},
  note         = {Machine review of arXiv:1908.02317}
}

read the original abstract

We study two functors between (partially) wrapped Fukaya categories. The first is the Orlov functor from the Fukaya category of a stop to the Fukaya category of the ambient sector. We give a geometric criterion for when this functor is spherical in the sense of Anno-Logvinenko. This criterion is a generalization of the situation where the stop comes from a Landau-Ginzburg model. The second functor is the Viterbo transfer map from a Liouville domain to a subdomain. We show that when the domain and subdomain are independently Weinstein, this functor is a homological epimorphism, which means that it becomes a localization after passing to module categories. This should be compared with a result of Ganatra-Pardon-Shende, which states that the Viterbo map is a genuine localization when the cobordism is Weinstein.

Figures

Figures reproduced from arXiv: 1908.02317 by the authors.

**Figure 1.** ij and ij+1 become isotopic up to shift after removing the extra stops. 3.2. An sectors. Returning to symplectic geometry, we consider a class of generalized stabilizations F¯hni = [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

**Figure 2.** Grothendieck construction on the left, mutated diagram on the right. The arrows represent Reeb flow. Explicitly, the semiorthogonal gluing of (3.2) has the form 7→7→ C = [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** The forward stopping manifold for iM. On diagonal morphisms, it is given by those same functors. On off-diagonal morphisms, it is induced by the map of (W(F¯), W(M))-bimodules (3.3) Γ(ıσM ) = (ıσM ,Id)∗∆W(M) ((iM)∗ ◦ ıσM ,(iM)∗) ∗ ∆ W(M 7→7→ σ ∪ M σN N) (iF¯h3i )∗ ◦ ı2[−1],(iM)∗ ∗ ∆ W(M 7→7→ σ ∪ M σN N) (iF¯h3i )∗ ◦ tw (ı0 → ı1 → ı2) [−1],(iM)∗ ∗ ∆ W(M 7→7→ σ ∪ M σN N) (iM)∗ Prop 2.10 ∼= and similarly for Γ† (… view at source ↗

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Abouzaid and S

M. Abouzaid and S. Ganatra. in preparation

work page

[2] [2]

Abouzaid and P

M. Abouzaid and P. Seidel. An open string analogue of Viterbo functoriality. Geom. Topol., 14:627–718, 2010

work page 2010

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Mohammed Abouzaid. A geometric criterion for generating the Fukaya category.Publ. Math. Inst. Hautes ´Etudes Sci., (112):191–240, 2010

work page 2010

[4] [4]

Spherical DG-functors

Rina Anno and Timothy Logvinenko. Spherical DG-functors. Journal of the European Mathematical Society , 19(9):2577–2656, 2017

work page 2017

[5] [5]

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work page 1990

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Nathaniel Bottman. Moduli spaces of witch curves topologically realize the 2-associahedra, 2017. arXiv:1712.01194

work page Pith review arXiv 2017

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