Fukaya categories of Coulomb branches as unique deformations

Jasper van de Kreeke

arxiv: 2607.00146 · v1 · pith:4XII7VURnew · submitted 2026-06-30 · 🧮 math.RT · math.RA· math.SG

Fukaya categories of Coulomb branches as unique deformations

Jasper van de Kreeke This is my paper

Pith reviewed 2026-07-02 00:56 UTC · model grok-4.3

classification 🧮 math.RT math.RAmath.SG

keywords Fukaya categoriesCoulomb branchesNilHecke algebrahorizontal Hilbert schemesrelative Fukaya categoriesdeformation theoryZ^2-gradingmatter divisor

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The pith

The Fukaya category of the horizontal Hilbert scheme is the unique Z^2-graded deformation of the NilHecke algebra category after removing the matter divisor.

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

The paper establishes that the Fukaya category of the horizontal Hilbert scheme, after removing its matter divisor, is the unique deformation of the category associated with the NilHecke algebra that preserves a compatible Z^2-grading. This circumvents the difficulty of directly determining Fukaya categories on Coulomb branches by first considering the complement of a divisor and then solving the resulting deformation problem. An existing identification of the endomorphism algebra of a generating Lagrangian with the NilHecke algebra is used as the starting point. Solving this graded deformation problem recovers the Fukaya category and confirms earlier results.

Core claim

After removing the matter divisor, the Fukaya category of the horizontal Hilbert scheme is the unique deformation of the NilHecke algebra category that admits a compatible Z^2-grading. The solution to the deformation problem yields the desired category.

What carries the argument

The Z^2-graded deformation of the NilHecke algebra category arising from the relative Fukaya category construction on the complement of the matter divisor.

If this is right

The Fukaya category can be determined by solving a deformation problem rather than direct geometric computation.
The result matches the one obtained by Aganagic et al.
Additional Z^2-grading restricts the possible deformations to a unique one.
This approach applies to other Coulomb branches where the matter divisor can be removed.

Where Pith is reading between the lines

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

This indicates that grading constraints may uniquely determine Fukaya categories in other relative settings.
The technique could be tested on vertical Hilbert schemes or different choices of divisors.
Connections to representation theory might be strengthened by viewing the NilHecke algebra as the base for these deformations.

Load-bearing premise

The endomorphism algebra of the chosen generating Lagrangian after removal of the matter divisor is exactly the NilHecke algebra, and this identification survives the deformation process without additional relations imposed by the geometry.

What would settle it

A calculation showing that the deformed algebra acquires extra relations beyond those of the NilHecke algebra, or the discovery of another Z^2-graded deformation that does not correspond to the Fukaya category of the Hilbert scheme.

Figures

Figures reproduced from arXiv: 2607.00146 by Jasper van de Kreeke.

**Figure 2.1.** Figure 2.1: This figure depicts the attachment of a chimney around a stop. The result is a Liouville [PITH_FULL_IMAGE:figures/full_fig_p015_2_1.png] view at source ↗

**Figure 3.1.** Figure 3.1: The simplified model of YO,W is given as a symmetric product of k base rectangles and k fiber cylinders. Each of the base rectangles has two stops whereas the fiber cylinders have one stop. The stops of the fiber cylinders are located at zero polar angle and are depicted with a slight offset to make space for later visual convenience. 3.3 Comparison of Liouville structures The functionality of the two Li… view at source ↗

**Figure 3.2.** Figure 3.2: This figure depicts the standard Lagrangian [PITH_FULL_IMAGE:figures/full_fig_p022_3_2.png] view at source ↗

**Figure 3.3.** Figure 3.3: This figure illustrates how to interpret elements of the NilHecke algebra as strand diagrams. It [PITH_FULL_IMAGE:figures/full_fig_p023_3_3.png] view at source ↗

**Figure 3.4.** Figure 3.4: This figure depicts the correspondence between the generators [PITH_FULL_IMAGE:figures/full_fig_p023_3_4.png] view at source ↗

**Figure 4.1.** Figure 4.1: The degree of the standard morphisms p = Xi , σi ∈ L ∩ L (1) is the turning number of the functions fq and fℏ when moving on the composite path from the identity along L (1) to p and back along L. Since the Lagrangians are symmetric products themselves, the composite path is drawn as two individual paths. In the case of σi , the difference yi −yi+1 makes a 180 degrees positive turn around zero, therefore… view at source ↗

read the original abstract

The symplectic geometry of Coulomb branches is complicated and it is particularly difficult to determine their Fukaya categories. Relative Fukaya categories present an approach to circumvent these difficulties by first computing the Fukaya category of the complement of a divisor and then solving a deformation problem. In this paper, we apply this approach to the specific case of horizontal Hilbert schemes by removing their matter divisor and narrowing down the set of possible deformations through an additional $ \mathbb{Z}^2 $-grading. We utilize an existing description of the Fukaya category after removal of the matter divisor, in particular we use a specific generating Lagrangian and the identification between its endomorphism algebra and the NilHecke algebra. The core of this paper consists of solving the deformation problem, after which we recover the result of Aganagic et al.

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 carries out the Z^2-graded deformation calculation for the Fukaya category of horizontal Hilbert schemes and recovers the Aganagic result, but it rests on an unverified algebra identification that may pick up extra relations.

read the letter

The paper solves the deformation problem for the Fukaya category of horizontal Hilbert schemes after removing the matter divisor. It takes an existing generating Lagrangian, identifies its endomorphism algebra with the NilHecke algebra, and uses the extra Z^2-grading to select the unique deformation that recovers the full category, matching the earlier result of Aganagic et al.

This is a concrete application of the relative Fukaya category method to a specific family. The grading step is a straightforward way to cut the deformation space down to one possibility, and working out the equations for horizontal Hilbert schemes adds a worked example that was not in the prior literature.

The soft spot is the reliance on the NilHecke identification. The paper invokes an existing description rather than re-deriving it or checking whether the ambient geometry imposes additional relations once the divisor is gone and the deformation parameter is turned on. If those relations appear, the deformation space shrinks and the uniqueness claim does not hold. The abstract gives no explicit deformation equations or bounds, so this part needs direct verification.

The work is for people already inside the Fukaya category and Coulomb branch literature who want to see the relative method applied to this case. A reader who knows the NilHecke algebra and the prior Aganagic result will follow it without trouble. It deserves peer review because the computation is specific and the method is checkable, even if the overall advance is incremental rather than foundational.

I would bring it to a reading group to examine the deformation equations. I would not cite it myself. Send it to referees.

Referee Report

2 major / 2 minor

Summary. The paper claims that after removing the matter divisor from the horizontal Hilbert scheme of a Coulomb branch, the Fukaya category is recovered as the unique deformation of the NilHecke algebra category that admits a compatible ℤ²-grading. It invokes an existing description of the post-removal Fukaya category (including a specific generating Lagrangian whose endomorphism algebra is identified with the NilHecke algebra) and solves the resulting deformation problem to match the result of Aganagic et al.

Significance. If the uniqueness argument holds, the work supplies a deformation-theoretic route to Fukaya categories of Coulomb branches that bypasses direct symplectic computations by combining relative Fukaya categories with an auxiliary grading. The approach explicitly builds on prior identifications rather than re-deriving them, which is a methodological strength when the deformation step is independently verified.

major comments (2)

[Abstract and core deformation section] Abstract and core deformation section: the uniqueness claim under the ℤ²-grading requires that the endomorphism algebra of the chosen generating Lagrangian equals the NilHecke algebra exactly after divisor removal and that this identification persists without extra relations imposed by the ambient symplectic geometry once the deformation parameter is turned on. The manuscript invokes the existing description for the identification but provides no derivation, bound, or check excluding geometry-induced relations that would shrink the deformation space and invalidate uniqueness.
[Deformation problem setup] Deformation problem setup: the argument that the ℤ²-grading narrows the deformation space to a single solution assumes the input category after divisor removal is precisely the one whose endomorphisms are the NilHecke algebra; if the geometric realization already encodes additional structure, the deformation step reduces to a consistency verification rather than an independent determination of the Fukaya category.

minor comments (2)

Notation for the ℤ²-grading and its compatibility with the deformation parameter should be introduced with an explicit definition or reference to the grading on the NilHecke algebra.
The manuscript should clarify in the introduction whether the recovered result of Aganagic et al. is obtained by direct comparison of generators and relations or by an abstract uniqueness argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We respond point by point to the major comments, clarifying the scope and reliance on prior work as stated in the manuscript.

read point-by-point responses

Referee: [Abstract and core deformation section] Abstract and core deformation section: the uniqueness claim under the ℤ²-grading requires that the endomorphism algebra of the chosen generating Lagrangian equals the NilHecke algebra exactly after divisor removal and that this identification persists without extra relations imposed by the ambient symplectic geometry once the deformation parameter is turned on. The manuscript invokes the existing description for the identification but provides no derivation, bound, or check excluding geometry-induced relations that would shrink the deformation space and invalidate uniqueness.

Authors: The manuscript explicitly invokes and relies upon an existing description (as stated in the abstract and introduction) for the Fukaya category after divisor removal, including the identification of the endomorphism algebra of the generating Lagrangian with the NilHecke algebra. The core contribution is the solution of the resulting deformation problem under the additional ℤ²-grading, which yields a unique deformation matching the result of Aganagic et al. We do not re-derive the input identification or provide an independent bound excluding further geometric relations; any such relations would already be incorporated (or excluded) in the cited prior description of the post-removal category. We can add a brief clarifying sentence in the introduction emphasizing this dependence on the existing identification. revision: partial
Referee: [Deformation problem setup] Deformation problem setup: the argument that the ℤ²-grading narrows the deformation space to a single solution assumes the input category after divisor removal is precisely the one whose endomorphisms are the NilHecke algebra; if the geometric realization already encodes additional structure, the deformation step reduces to a consistency verification rather than an independent determination of the Fukaya category.

Authors: The paper's approach is precisely to take the post-removal category as given by the existing description (with endomorphism algebra the NilHecke algebra) and then apply the ℤ²-grading to narrow the deformation space to a unique solution. This is presented as a methodological route that combines relative Fukaya categories with an auxiliary grading, rather than an independent symplectic computation from scratch. The deformation analysis is therefore conditional on the input category; if additional structure were present it would modify the starting point, but the uniqueness result holds for the category as described in the cited work. revision: no

Circularity Check

0 steps flagged

No significant circularity: core deformation solve is independent

full rationale

The paper takes the post-divisor-removal Fukaya category and its NilHecke endomorphism identification as an external input (explicitly 'we utilize an existing description'), then performs an independent deformation calculation under the additional Z^2-grading to produce a unique candidate. This calculation does not redefine the input algebra in terms of the output or force the result by construction; it solves a standard deformation problem whose solution is then matched to the known target. No self-citation chain is load-bearing for the uniqueness claim, and the identification is treated as given rather than derived inside the paper. The derivation therefore remains self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Fukaya categories, the definition of the NilHecke algebra, and the existence of a Z^2-grading compatible with the deformation; no free parameters or invented entities are apparent from the abstract.

axioms (2)

domain assumption The Fukaya category of the complement of the matter divisor is generated by a Lagrangian whose endomorphism algebra is the NilHecke algebra.
Invoked when the paper states it utilizes an existing description after removal of the matter divisor.
domain assumption Deformations of the endomorphism algebra that preserve the Z^2-grading correspond exactly to deformations arising from the original symplectic manifold.
This is the load-bearing step that allows the uniqueness argument to recover the full Fukaya category.

pith-pipeline@v0.9.1-grok · 5657 in / 1474 out tokens · 24204 ms · 2026-07-02T00:56:53.822971+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 10 canonical work pages · 1 internal anchor

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Paul Seidel. “FukayaA ∞-structures associated to Lefschetz fibrations. I”. In:J. Symplectic Geom.10.3 (2012), pp. 325–388.issn: 1527-5256,1540-2347.doi:10.4310/jsg.2012.v10.n3.a1. url:https://doi-org.libproxy.berkeley.edu/10.4310/jsg.2012.v10.n3.a1

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Paul Seidel.Fukaya categories and deformations. 2002. arXiv:math/0206155 [math.SG].url: https://arxiv.org/abs/math/0206155

work page internal anchor Pith review Pith/arXiv arXiv 2002
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On partially wrapped Fukaya categories

Zachary Sylvan. “On partially wrapped Fukaya categories”. In:Journal of Topology12.2 (Jan. 2019), pp. 372–441.issn: 1753-8424.doi:10.1112/topo.12088.url:http://dx.doi.org/10. 1112/topo.12088

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Ben Webster.Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems. 2024. arXiv:1905.04623 [math.AG].url:https://arxiv.org/abs/1905. 04623

work page arXiv 2024
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work page arXiv 2024

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Homological knot invariants from mirror symmetry

Mina Aganagic. “Homological knot invariants from mirror symmetry”. eng. In:EMS Press eBooks. Vol. 3. Str. des 17. Juni 136, 10623 Berlin, Germany: EMS Press, 2023, pp. 2108–2144.isbn: 9783985470617

2023

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Mina Aganagic et al.Quiver Hecke algebras from Floer homology in Couloumb branches

[3] [3]

arXiv:2406.04258 [math.SG].url:https://arxiv.org/abs/2406.04258

work page arXiv

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Severin Barmeier and Zhengfang Wang.Deformations of path algebras of quivers with re- lations. 2023. arXiv:2002.10001 [math.QA].url:https://arxiv.org/abs/2002.10001

work page arXiv 2023

[5] [5]

London Mathematical Society student texts ; 99

Raf Bocklandt.A gentle introduction to homological mirror symmetry / Raf Bocklandt, University of Amsterdam.eng. London Mathematical Society student texts ; 99. Cambridge: Cambridge University Press, 2021.isbn: 1-108-69245-1

2021

[6] [6]

Noncommutative mirror symmetry for punctured surfaces

Raf Bocklandt. “Noncommutative mirror symmetry for punctured surfaces”. In:Trans. Amer. Math. Soc.368.1 (2016). With an appendix by Mohammed Abouzaid, pp. 429–469

2016

[7] [7]

Deformed mirror symmetry for punctured surfaces

Raf Bocklandt and Jasper van de Kreeke. “Deformed mirror symmetry for punctured surfaces”. In:Journal of Algebra672 (2025), pp. 413–602.issn: 0021-8693.doi:https://doi.org/10. 1016/j.jalgebra.2025.02.024.url:https://www.sciencedirect.com/science/article/ pii/S0021869325000997. 35

2025

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Towards a mathematical def- inition of Coulomb branches of 3-dimensional N=4 gauge theories, II

Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. “Towards a mathematical def- inition of Coulomb branches of 3-dimensional N=4 gauge theories, II”. eng. In:Advances in theoretical and mathematical physics22.5 (2018), pp. 1071–1147.issn: 1095-0761

2018

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Cosilting complexes and AIR-cotilting mod- ules

Sergio Chouhy and Andrea Solotar. “Projective resolutions of associative algebras and ambiguities”. In:J. Algebra432 (2015), pp. 22–61.issn: 0021-8693,1090-266X.doi:10.1016/j.jalgebra. 2015.02.019.url:https://doi-org.libproxy.berkeley.edu/10.1016/j.jalgebra.2015.02. 019

work page doi:10.1016/j.jalgebra 2015

[10] [10]

Duality between Lagrangian and Legendrian invariants

Tobias Ekholm and Yankı Lekili. “Duality between Lagrangian and Legendrian invariants”. In: Geometry & Topology27.6 (Aug. 2023), pp. 2049–2179.issn: 1465-3060.doi:10.2140/gt. 2023.27.2049.url:http://dx.doi.org/10.2140/gt.2023.27.2049

work page doi:10.2140/gt 2023

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Weinstein manifolds revisited

Yakov Eliashberg et al. “Weinstein manifolds revisited”. eng. In:Modern Geometry. Vol. 99. Proceedings of Symposia in Pure Mathematics. Providence, Rhode Island: American Mathematical Society, 2018, pp. 59–82.isbn: 9781470440947

2018

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Covariantly functorial wrapped Floer theory on Liouville sectors

Sheel Ganatra, John Pardon, and Vivek Shende. “Covariantly functorial wrapped Floer theory on Liouville sectors”. eng. In:Publications math´ ematiques de l’IH´ES131.1 (2020), pp. 73–200. issn: 0073-8301

2020

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Sectorial descent for wrapped Fukaya categories

Sheel Ganatra, John Pardon, and Vivek Shende. “Sectorial descent for wrapped Fukaya categories”. eng. In:Journal of the American Mathematical Society37.2 (2023), pp. 499–635.issn: 0894- 0347

2023

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Mirror symmetry in three-dimensional gauge theories

Nathan Seiberg Ken Intriligator. “Mirror symmetry in three-dimensional gauge theories”. In:Phys- ical Letters B387.3 (1996), pp. 513–519

1996

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ox.ac.uk/beem/papers/kontsevich_soibelman_deformation_theory_1.pdf

Maxim Kontsevich and Yan Soibelman.Deformation Theory I.url:https://people.maths. ox.ac.uk/beem/papers/kontsevich_soibelman_deformation_theory_1.pdf

[16] [16]

Jasper van de Kreeke.A ∞-Deformations and their Derived Categories. 2023. arXiv:2308. 08026 [math.KT].url:https://arxiv.org/abs/2308.08026

work page arXiv 2023

[17] [17]

FukayaA ∞-structures associated to Lefschetz fibrations. I

Paul Seidel. “FukayaA ∞-structures associated to Lefschetz fibrations. I”. In:J. Symplectic Geom.10.3 (2012), pp. 325–388.issn: 1527-5256,1540-2347.doi:10.4310/jsg.2012.v10.n3.a1. url:https://doi-org.libproxy.berkeley.edu/10.4310/jsg.2012.v10.n3.a1

work page doi:10.4310/jsg.2012.v10.n3.a1 2012

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Paul Seidel.Fukaya categories and deformations. 2002. arXiv:math/0206155 [math.SG].url: https://arxiv.org/abs/math/0206155

work page internal anchor Pith review Pith/arXiv arXiv 2002

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On partially wrapped Fukaya categories

Zachary Sylvan. “On partially wrapped Fukaya categories”. In:Journal of Topology12.2 (Jan. 2019), pp. 372–441.issn: 1753-8424.doi:10.1112/topo.12088.url:http://dx.doi.org/10. 1112/topo.12088

work page doi:10.1112/topo.12088.url:http://dx.doi.org/10 2019

[20] [20]

Ben Webster.Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems. 2024. arXiv:1905.04623 [math.AG].url:https://arxiv.org/abs/1905. 04623

work page arXiv 2024

[21] [21]

Ben Webster.Coherent sheaves and quantum Coulomb branches II: quiver gauge theo- ries and knot homology. 2024. arXiv:2211.02099 [math.AG].url:https://arxiv.org/abs/ 2211.02099. 36

work page arXiv 2024