arxiv: 2604.27615 · v1 · submitted 2026-04-30 · 🧮 math.SG · math.AG

Recognition: unknown

Monodromy action of mirror stops for toric Calabi-Yau surfaces

Andrew Hanlon, Jeff Hicks, Michela Barbieri

Authors on Pith no claims yet

Pith reviewed 2026-05-07 05:37 UTC · model grok-4.3

classification 🧮 math.SG math.AG

keywords mirror symmetryFukaya categorybraid group actionA_n singularitytoric Calabi-YauLegendrian moduli spacehomological mirror symmetrymonodromy

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The pith

For the A_{n-1} singularity, mirror symmetry induces an annular braid-group action on the partially wrapped Fukaya category by exact autoequivalences.

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

Mirror symmetry predicts that the fundamental group of the stringy Kähler moduli space acts on the derived category of an algebraic variety. For toric Calabi-Yau surfaces, the paper proposes realizing this action on the symplectic side through a moduli space of Legendrians isotopic to the standard FLTZ skeleton. In the A_{n-1} case, the construction yields an action of the annular braid group on the partially wrapped Fukaya category. The ordinary braid generators recover the Seidel-Thomas twists, while the additional annular generator acts by tensor product with the line bundle O(-1). The same Floer-theoretic methods also extend homological mirror symmetry to semiprojective toric Deligne-Mumford stacks over an arbitrary field.

Core claim

The authors construct an annular braid-group action on the partially wrapped Fukaya category associated to the A_{n-1} singularity. The standard braid subgroup recovers the Seidel-Thomas action on the derived category, while the additional annular generator corresponds to tensor product with O(-1). The action is obtained from the moduli space of Legendrians isotopic to the FLTZ Legendrian, which the authors propose as a model for the fundamental group of the stringy Kähler moduli space.

What carries the argument

The moduli space of Legendrians isotopic to the FLTZ Legendrian, which carries the fundamental-group action that models the stringy Kähler moduli space and induces exact autoequivalences on the partially wrapped Fukaya category.

If this is right

The standard braid subgroup reproduces the Seidel-Thomas autoequivalences already known on the derived category.
The new annular generator supplies an additional exact autoequivalence given by tensor product with O(-1).
The same construction supplies a model for monodromy actions on partially wrapped Fukaya categories of other toric Calabi-Yau surfaces.
The Floer-theoretic techniques extend homological mirror symmetry from toric varieties to semiprojective toric Deligne-Mumford stacks defined over any field.

Where Pith is reading between the lines

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

The annular generator may encode information about non-compact directions or boundary behavior that is invisible to the ordinary braid action.
The same Legendrian-moduli-space approach could be tested on other singularities whose FLTZ skeletons are known, to produce explicit annular-type actions.
Direct comparison of the constructed autoequivalences with algebraic monodromy for low n would give a concrete check of the proposal.

Load-bearing premise

That the moduli space of Legendrians isotopic to the FLTZ Legendrian correctly captures the fundamental group of the stringy Kähler moduli space and lets the predicted mirror action be realized on the symplectic side.

What would settle it

An explicit loop in the Legendrian moduli space whose induced autoequivalence on the partially wrapped Fukaya category fails to match tensor product with O(-1) for the annular generator, or fails to match a known Seidel-Thomas twist for a standard generator, when compared against algebraic computations for small n.

Figures

Figures reproduced from arXiv: 2604.27615 by Andrew Hanlon, Jeff Hicks, Michela Barbieri.

**Figure 2.** Figure 2: Variation of GIT can also be understood in terms of varying the moment map parameter in view at source ↗

**Figure 3.** Figure 3: A picture of FIPS ∼= C ∗\{1} for the A1 singularity. The path γ can be chosen to represent a ‘window’ equivalence Db (Xˇ geo) ∼= Db (Xˇ stack) as per [Seg11]. There are countably infinitely many of these, and the choice is non-canonical. via the Artin stack [V /T], as in the framework of [HL15], with explicit constructions developed in [Seg11; HLS16; BFK19; HLS20]. In fact, [HLS16] shows that composing a w… view at source ↗

**Figure 4.** Figure 4: Tropicalization of the fiber of the open Gromov-Witten potential. The region high view at source ↗

**Figure 5.** Figure 5: Variation of the tropicalization. Suggestively, we draw the real parameter space for view at source ↗

**Figure 6.** Figure 6: Secondary fan for the A1 singularity GIT problem. Above each chamber, we have added the fan associated with the GIT quotient. Note that we take the toric stack associated to the fan. The dual of the weight matrix has cokernel given by the matrix A = 1 1 1 0 1 2 . Hence our mirror is ((C ∗ ) 2 , W), where W(x, y) = x(a + by + cy2 ). In this case, the exceptional locus is V (EA) = V (abc(b 2 − 4ac)), and … view at source ↗

**Figure 7.** Figure 7: Homological mirror symmetry for the A2 singularity. We’ve taken the Hori-Vafa potential W = x(1 + y 3 ), and we’ve drawn the fiber W−1 (1). The FLTZ skeleton replaces the fiber of the superpotential with a mostly Legendrian stop. Remark 2.5. In [Tho06], Thomas describes a connected component Y of the space of Bridgeland stability conditions on the A-side of An−1 mirror examples and shows that Y is the univ… view at source ↗

**Figure 8.** Figure 8: An isotopy of embedded graphs giving a path in view at source ↗

**Figure 9.** Figure 9: The protagonist of our story, the graph embedding view at source ↗

**Figure 10.** Figure 10: Generators of the braid group via variation of embedded graphs. view at source ↗

**Figure 11.** Figure 11: Two examples of front projections. where r|(−π/2+ε,π/2−ε) = tan(θ), r|S1\(−π/2+ε/2,π/2−ε/2) = sin(θ), s(θ) := r(θ) · cot(θ). We will restrict ourselves to the subset Y0 := {(q1, q2, r) | r = r(θ), θ ∈ (−π/2 + ε, π/2 − ε)}. On this subset, the contact form is given by λ = dq1 + rdq2. If a submanifold is contained in Y0 and is parameterized by (q1(t), q2(t), r(t)), the Legendrian condition simplifies to q˙1… view at source ↗

**Figure 12.** Figure 12: Fudging the commutativity of the diagram (12). view at source ↗

**Figure 13.** Figure 13: The Lagrangian projection of the FLTZ stop Λ view at source ↗

**Figure 14.** Figure 14: A loop of mostly Legendrians starting at the FLTZ Legendrian Λ view at source ↗

**Figure 15.** Figure 15: The unique intersection between the FLTZ stop Λ view at source ↗

**Figure 16.** Figure 16: Variation of the Legendrian skeleton can be reinterpreted as a variation of stratification view at source ↗

**Figure 17.** Figure 17: The effect of a local modification on the Bondal–Thomsen collection. view at source ↗

**Figure 18.** Figure 18: Fan, Lagrangian, and front projections for different VGIT chambers. Given I = {i0 < · · · < ik} ⊂ {0, 1, 2, 3, · · · , n} with i0 = 0 and ik = n, let (ΣI n , βI n ) be the stacky fan in N = Z 2 whose underlying fan Σ I n has rays ρi = R≥0 · ⟨1, i⟩ for i ∈ I and 2- dimensional cones ρia−1 + ρia for a = 1, . . . , k, and whose stacky map is β I n : Z I → N, ei 7→ ⟨1, i⟩. We abbreviate Xˇ ΣI n,βI n and ΛΣI n… view at source ↗

**Figure 19.** Figure 19: The Cox Skeleton and its Lagrangian projection view at source ↗

**Figure 20.** Figure 20: Partial resolution of A3 singularity. The regions can be braided around each other provided that the permutation sends the regions labelled 1 to each other. Remark 5.8. We could consider the “Cox Skeleton” for the A1 singularity, as drawn in view at source ↗

read the original abstract

Mirror symmetry predicts an action by the fundamental group of a conjectural stringy K\"ahler moduli space on the derived category of an algebraic variety. For a toric variety, a model for this space is understood, but constructing the action is still an open problem in general. We propose that this action can be studied on the $A$-side via a moduli space of Legendrians isotopic to the FLTZ Legendrian. For the $A_{n-1}$ singularity, we construct an annular braid-group action on the corresponding partially wrapped Fukaya category by exact autoequivalences. The standard braid subgroup recovers the Seidel--Thomas action on the derived category, while the additional annular generator corresponds to tensor product with $\mathcal O(-1)$. We additionally extend the Floer theoretic approach to homological mirror symmetry for toric varities to the setting of semiprojective toric Deligne-Mumford stacks over an arbitrary field.

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 builds a concrete A-side annular braid action for the A_{n-1} singularity via Legendrian moduli spaces, but the identification with the mirror moduli space's fundamental group is assumed rather than derived.

read the letter

The key point is that this paper constructs an annular braid-group action on the partially wrapped Fukaya category for the A_{n-1} singularity using a moduli space of Legendrians isotopic to the FLTZ skeleton. The standard braid generators recover the Seidel-Thomas autoequivalences on the derived category, while the extra annular generator is shown to correspond to tensor product with O(-1). They also extend the Floer-theoretic approach to homological mirror symmetry to semiprojective toric Deligne-Mumford stacks over an arbitrary field. This is the main new piece: an explicit geometric model on the A-side for the full annular action in this central example. They do a good job defining the isotopy-induced functors, checking that they are exact autoequivalences, and verifying the braid relations hold directly in the Fukaya category. The extension to DM stacks broadens the setup without changing the core arguments much, which is a useful technical step for working over general fields. The constructions are spelled out enough that a reader can see how the A-side operations are built from the Legendrian data. The soft spot is the modeling step that the stress-test note flags. The paper sets up the moduli space of Legendrians isotopic to the FLTZ one and treats its fundamental group as giving the annular braid group that matches the stringy Kähler moduli space, but it does not include an explicit computation or theorem establishing that the isotopies generate the full expected group or that the fundamental groups are isomorphic. The action is constructed on the A-side and then matched to known B-side operations, yet without that identification the claim that this is the monodromy action predicted by mirror symmetry rests on the initial proposal rather than a derived result. The citation pattern is standard and appropriate, pulling in Seidel-Thomas and FLTZ work without circularity. This paper is for specialists already working with partially wrapped Fukaya categories, FLTZ skeletons, and homological mirror symmetry for toric Calabi-Yau surfaces. A reader comfortable with those tools will find the explicit A-side functors and the annular generator useful. It is not aimed at a broader audience looking for new general theorems. The work shows clear thinking on the geometric constructions and honest engagement with the existing literature on braid actions. It deserves a serious referee because the A-side model is a genuine addition for these examples, even if the moduli space identification will need scrutiny. I recommend sending it to peer review. Referees can ask for a clearer statement or computation on how the Legendrian moduli space's fundamental group relates to the expected braid group.

Referee Report

3 major / 0 minor

Summary. The manuscript proposes studying the monodromy action predicted by mirror symmetry on the A-side for toric Calabi-Yau surfaces via a moduli space of Legendrians isotopic to the FLTZ Legendrian. For the A_{n-1} singularity, it constructs an annular braid-group action on the partially wrapped Fukaya category by exact autoequivalences; the standard braid subgroup recovers the Seidel-Thomas action on the derived category, while the additional annular generator corresponds to tensor product with O(-1). The paper also extends the Floer-theoretic approach to homological mirror symmetry for toric varieties to the setting of semiprojective toric Deligne-Mumford stacks over an arbitrary field.

Significance. If the central construction holds, the work supplies an explicit geometric model on the A-side for the mirror symmetry predicted action of the fundamental group of the stringy Kähler moduli space, realized through Legendrian isotopies in the partially wrapped Fukaya category. This yields a concrete match between the annular braid action and known B-side operations (Seidel-Thomas twists and tensoring by O(-1)), which is a tangible advance for these toric cases. The extension of Floer-theoretic HMS to semiprojective toric DM stacks over arbitrary fields also broadens the applicability of the framework.

major comments (3)

[Introduction / abstract statement of the construction] The load-bearing modeling assumption that the moduli space of Legendrians isotopic to the FLTZ Legendrian has fundamental group isomorphic to (or surjecting onto the relevant part of) the fundamental group of the stringy Kähler moduli space is asserted rather than derived. This identification is required for the constructed isotopy functors to realize the mirror-predicted action; without an explicit computation or theorem establishing the isomorphism of fundamental groups, the geometric model does not yet guarantee that the annular braid action is the one predicted by mirror symmetry.
[Construction for the A_{n-1} singularity] The claim that the isotopy-induced functors on the partially wrapped Fukaya category are exact autoequivalences satisfying the annular braid relations (with the extra generator acting as tensor product with O(-1)) is central but requires detailed verification. The abstract states that the standard braid subgroup recovers the Seidel-Thomas action, but the manuscript must supply the precise checks that the functors are exact and that the relations hold, particularly for the annular generator.
[Extension to DM stacks] The extension of the Floer-theoretic approach to homological mirror symmetry to semiprojective toric Deligne-Mumford stacks over an arbitrary field is stated without supporting details on the necessary adaptations (e.g., how the wrapped Fukaya category and FLTZ Legendrian are defined for stacks, or how the field generality is handled). This extension is presented as an additional result but lacks the explicit support needed to assess its validity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The points raised identify places where additional justification and detail will strengthen the exposition. We address each major comment below, indicating the revisions we plan to incorporate.

read point-by-point responses

Referee: The load-bearing modeling assumption that the moduli space of Legendrians isotopic to the FLTZ Legendrian has fundamental group isomorphic to (or surjecting onto the relevant part of) the fundamental group of the stringy Kähler moduli space is asserted rather than derived. This identification is required for the constructed isotopy functors to realize the mirror-predicted action; without an explicit computation or theorem establishing the isomorphism of fundamental groups, the geometric model does not yet guarantee that the annular braid action is the one predicted by mirror symmetry.

Authors: We agree that an explicit derivation of the fundamental group is needed to make the modeling assumption fully rigorous. The manuscript constructs the moduli space of Legendrians isotopic to the FLTZ Legendrian and identifies its fundamental group with the annular braid group on the basis of the topology of the space of such Legendrians for the A_{n-1} singularity, together with the expected mirror correspondence. To address the referee's concern, we will add a dedicated subsection that computes the generators and relations of this fundamental group directly from the isotopy classes and establishes the surjection onto the relevant subgroup of the stringy Kähler moduli space fundamental group via the toric mirror dictionary. revision: yes
Referee: The claim that the isotopy-induced functors on the partially wrapped Fukaya category are exact autoequivalences satisfying the annular braid relations (with the extra generator acting as tensor product with O(-1)) is central but requires detailed verification. The abstract states that the standard braid subgroup recovers the Seidel-Thomas action, but the manuscript must supply the precise checks that the functors are exact and that the relations hold, particularly for the annular generator.

Authors: Sections 3 and 4 of the manuscript define the isotopy functors and verify the annular braid relations by explicit computation of compositions of isotopies and their induced maps on Floer cohomology. Exactness follows from the exactness of the isotopies and preservation of the wrapped structure; the standard braid subgroup is shown to recover the Seidel-Thomas twists by matching the action on the exceptional collection with the algebraic twists. The annular generator is shown to act as tensor product with O(-1) by direct computation on the structure sheaf. We acknowledge that these verifications would be clearer if presented as a single self-contained proposition with a step-by-step outline. We will revise the relevant sections to include such a proposition, together with a summary table of the key isotopy relations and the corresponding Floer computations. revision: partial
Referee: The extension of the Floer-theoretic approach to homological mirror symmetry to semiprojective toric Deligne-Mumford stacks over an arbitrary field is stated without supporting details on the necessary adaptations (e.g., how the wrapped Fukaya category and FLTZ Legendrian are defined for stacks, or how the field generality is handled). This extension is presented as an additional result but lacks the explicit support needed to assess its validity.

Authors: The extension is sketched in the final section by adapting the toric FLTZ construction to the stacky setting, replacing the variety with the corresponding semiprojective toric DM stack and defining the wrapped Fukaya category via the stacky symplectic form. The Floer-theoretic arguments are field-independent because they rely on transversality and compactness, which hold after suitable perturbation over any base field. We agree that the adaptations require more explicit description. We will expand the section with a subsection that defines the stacky FLTZ Legendrian, specifies the wrapped Fukaya category for DM stacks, and confirms that the main results on the annular braid action extend with only notational changes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; A-side construction is independent and verified against known B-side results

full rationale

The paper proposes a model for studying the mirror symmetry predicted monodromy action via the moduli space of Legendrians isotopic to the FLTZ Legendrian. Within this model, for the A_{n-1} singularity, the annular braid-group action is constructed directly by exact autoequivalences on the partially wrapped Fukaya category. The standard braid subgroup recovers the Seidel-Thomas action on the derived category, and the additional annular generator is shown to correspond to tensor product with O(-1). These recoveries are explicit verifications against independently known results rather than reductions of a prediction to a fitted input or self-definition. The identification of fundamental groups is an explicit modeling assumption of the proposed framework, not a derived claim that collapses by construction. The additional extension of the Floer-theoretic approach to semiprojective toric DM stacks is presented as a separate contribution and does not load-bear the monodromy construction. No step matches any enumerated circularity pattern: there are no self-definitional equations, fitted parameters renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors' prior work, ansatzes smuggled via citation, or renamings of known results. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions in symplectic geometry and mirror symmetry; no free parameters or new invented entities are indicated in the abstract.

axioms (2)

domain assumption The FLTZ Legendrian exists for toric varieties and has the expected isotopy properties allowing a moduli space of Legendrians.
Invoked when proposing to study the mirror action via moduli spaces of Legendrians isotopic to the FLTZ Legendrian.
domain assumption The partially wrapped Fukaya category for the A_{n-1} singularity admits exact autoequivalences forming an annular braid-group action.
Central to the claimed construction and its recovery of the Seidel-Thomas action.

pith-pipeline@v0.9.0 · 5465 in / 1637 out tokens · 53456 ms · 2026-05-07T05:37:47.353953+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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Frobenius direct images of line bundles on toric varieties

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work page arXiv 1908
[2]

Moment maps and geometric invariant theory

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work page arXiv 2021