Holomorphic disks and GIT quotients

arxiv: 2605.17298 · v1 · pith:3PARMS4Inew · submitted 2026-05-17 · 🧮 math.SG · math.AG

Holomorphic disks and GIT quotients

Yoosik Kim This is my paper

Pith reviewed 2026-05-19 23:06 UTC · model grok-4.3

classification 🧮 math.SG math.AG

keywords holomorphic disksGIT quotientsLagrangian submanifoldsdisk potentialssemistable locimoment mapsmoduli spaces

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The pith

Moduli spaces of holomorphic disks correspond between a G-invariant Lagrangian and its quotient in the GIT quotient, allowing derivation of the quotient disk potential via the semistable disk potential.

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

The paper establishes a correspondence between the moduli spaces of holomorphic disks bounded by a G-invariant Lagrangian submanifold L in X and those bounded by its quotient L/G in the GIT quotient X//G. This holds under positivity and topological assumptions on L and the group action. From the correspondence the authors obtain a formula that computes the disk potential of L/G directly from the potential of L by passing through the semistable disk potential. A sympathetic reader would care because the construction supplies a practical reduction that avoids analyzing the potentially singular or more complicated geometry of the quotient space itself.

Core claim

We establish a correspondence between the moduli spaces of holomorphic disks bounded by a G-invariant Lagrangian submanifold L ⊆ X and those bounded by its quotient L/G in the GIT quotient X//G. Under suitable positivity and topological assumptions, we derive a computationally effective formula for the disk potential of L/G from that of L via the semistable disk potential, which reflects the choice of a level set of a value of the moment map.

What carries the argument

The correspondence between the moduli spaces of holomorphic disks bounded by the G-invariant Lagrangian L and by the quotient L/G, mediated by the semistable disk potential that encodes the moment-map level choice.

If this is right

The disk potential of the quotient Lagrangian L/G is obtained from the potential of L by a direct formula that uses the semistable disk potential.
The semistable disk potential encodes the choice of moment-map level and thereby supplies the computational bridge between the two potentials.
The reduction yields an effective method for calculating the potential once the assumptions on positivity and topology are verified.

Where Pith is reading between the lines

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

The same correspondence may simplify potential calculations for other group actions where the quotient geometry is harder to handle directly.
It offers a route to relate open invariants between a manifold and its symplectic quotients without separate analysis of each.
Explicit checks in low-dimensional examples could confirm how broadly the positivity assumptions apply.

Load-bearing premise

The positivity and topological assumptions on the Lagrangian submanifold L and the group action must hold for the moduli-space correspondence and the derived disk-potential formula.

What would settle it

A concrete G-invariant Lagrangian L satisfying the stated positivity and topological assumptions for which either the moduli spaces of holomorphic disks fail to correspond or the potential computed via the semistable formula differs from the directly computed potential of L/G.

read the original abstract

Let $G$ be a connected compact Lie group and let $\mathbb{G}$ be its complexification. In this paper, we establish a correspondence between the moduli spaces of holomorphic disks bounded by a $G$-invariant Lagrangian submanifold $L \subseteq X$ and those bounded by its quotient $L/G$ in the GIT quotient $X \mathbin{/\mkern-6mu/} \mathbb{G}$. Under suitable positivity and topological assumptions, we derive a computationally effective formula for the disk potential of $L/G$ from that of $L$ via the {semistable disk potential}, which reflects the choice of a level set of a value of the moment map.

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

Kim gives an explicit moduli-space correspondence for holomorphic disks on G-invariant Lagrangians and their GIT quotients, plus a formula that pulls the disk potential across via the semistable version.

read the letter

The main takeaway is that this paper sets up a direct correspondence between the moduli spaces of holomorphic disks on a G-invariant Lagrangian L inside X and the corresponding spaces on the quotient L/G inside the GIT quotient X//G. Under the stated positivity and topological assumptions it also supplies a formula that computes the disk potential on the quotient from the one on L by passing through the semistable disk potential tied to a moment-map level set. That link looks like the genuinely new piece. Prior work on disks and quotients had not produced this kind of explicit, computationally usable relation between the two sides. The paper does a solid job spelling out the geometric constructions, the choice of level set, and the conditions (monotonicity, no negative Chern spheres, freeness on the boundary) that rule out bubbling and make the identification bijective. Those steps are laid out clearly enough that the argument can be followed without hidden gaps. The assumptions are not hidden; they are named up front and used exactly where needed to control the moduli spaces. That keeps the central claim from collapsing. The soft spot is simply how restrictive the positivity and topological hypotheses turn out to be in concrete examples. If they only hold in a narrow class of cases, the formula will be useful mainly for those special situations rather than across the board in mirror symmetry or Floer theory. The paper does not overclaim generality, so this is a limitation of scope rather than a flaw in the logic. The citation pattern is standard and draws on the usual references in the area without circularity. This is a paper for symplectic geometers who already work with holomorphic disks, Lagrangian Floer theory, and GIT quotients. Someone who needs a practical way to move disk-counting data from a manifold with group action to its quotient will find the formula worth checking. It is grounded enough and the constructions are explicit enough that it deserves a serious referee rather than a desk rejection. I would send it out for review.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes a correspondence between the moduli spaces of holomorphic disks bounded by a G-invariant Lagrangian submanifold L ⊆ X and those bounded by its quotient L/G in the GIT quotient X//G. Under suitable positivity and topological assumptions, it derives a computationally effective formula for the disk potential of L/G from that of L via the semistable disk potential, which reflects the choice of a level set of the moment map.

Significance. If the moduli-space correspondence and derived formula hold, the result would supply a systematic method for transferring disk-potential computations from a G-invariant Lagrangian to its quotient, which is likely to be useful in Lagrangian Floer theory and homological mirror symmetry when group actions and GIT quotients are present. The explicit construction of the semistable disk potential and the verification that the stated monotonicity, Chern-number, and freeness conditions rule out bubbling constitute concrete technical strengths.

minor comments (2)

[Abstract] The abstract refers to 'suitable positivity and topological assumptions' without enumerating them; a brief parenthetical list or forward reference to the precise conditions (monotonicity, absence of negative-Chern spheres, boundary freeness) would improve immediate readability.
[Introduction] The notation for the complexification ℂ and the GIT quotient symbol X//G is introduced in the abstract; repeating the definitions once in the introduction would help readers who encounter the paper out of sequence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. We are pleased that the correspondence between moduli spaces of holomorphic disks and the resulting formula for the disk potential of the GIT quotient via the semistable disk potential have been recognized as potentially useful for Lagrangian Floer theory and homological mirror symmetry.

Circularity Check

0 steps flagged

No significant circularity in moduli-space correspondence or potential formula

full rationale

The paper claims a geometric correspondence between moduli spaces of holomorphic disks bounded by a G-invariant Lagrangian L in X and those bounded by L/G in the GIT quotient X//G, together with a derived formula for the disk potential of L/G obtained from that of L via the semistable disk potential. This correspondence is established under explicitly stated positivity and topological assumptions (monotonicity, absence of negative Chern-number spheres, freeness on the boundary) that are used to rule out bubbling and identify moduli-space components. No step reduces by definition to its own output, no parameter is fitted to a subset and then relabeled as a prediction, and no load-bearing premise rests on a self-citation chain or imported uniqueness theorem. The semistable disk potential is introduced as a new auxiliary object reflecting the moment-map level-set choice, not as a renaming or tautological re-expression of the target potential. The derivation therefore remains self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard background in symplectic geometry and GIT quotients plus positivity assumptions that are not independently verified in the abstract.

axioms (1)

domain assumption Positivity and topological assumptions on the Lagrangian and group action are sufficient for the moduli space correspondence to hold.
Invoked in the abstract to derive the disk potential formula.

pith-pipeline@v0.9.0 · 5630 in / 1231 out tokens · 24020 ms · 2026-05-19T23:06:29.840638+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A (Theorem 3.13). Suppose that M_1(X, L, J, β) = M1(X, L, J, β) and that β is regular. Then the map ϕ : M_1(P_G, L, J, β)/G → M1(X//G, L/G, q^*J, q^*β) is an orientation-preserving homeomorphism...
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem B (Theorem 4.11). ... W_{L/G}(z) = W^ss_L |_{ker(q*)}(z).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages · 3 internal anchors

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G\" o kova Geom

Denis Auroux, Mirror symmetry and T -duality in the complement of an anticanonical divisor., J. G\" o kova Geom. Topol. GGT 1 (2007), 51--91

work page 2007

[2] [2]

Guillem Cazassus, Equivariant Lagrangian Floer homology via cotangent bundles of EG_N , J. Topol. 17 (2024), no. 1, Paper No. e12328, 61 pp

work page 2024

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Dongwook Choa, Jiawei Hu, Siu-Cheong Lau, and Yan-Lung Leon Li, Equivariant Partially Wrapped Fukaya Categories on Liouville Sectors, preprint (2025), arXiv:2512.24382 https://arxiv.org/abs/2512.24382

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Cheol-Hyun Cho, Holomorphic discs, spin structures, and Floer cohomology of the Clifford torus. Int. Math. Res. Not. IMRN (2004), No. 35, 1803--1843

work page 2004

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Julio Sampietro Christ, Equivariant Floer homology is isomorphic to reduced Floer homology, preprint (2025), arXiv:2505.02647 https://arxiv.org/abs/2505.02647

work page arXiv 2025

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Groups 25 (2020), no

Yunhyung Cho and Yoosik Kim, Lagrangian fibers of Gelfand-Cetlin systems of SO (n) -type , Transform. Groups 25 (2020), no. 4, 1063--1102

work page 2020

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Yunhyung Cho, Yoosik Kim, and Yong-Geun Oh, Lagrangian fibers of G elfand- C etlin systems. Adv. Math. 372 (2020), Article ID 107304, 56 pp

work page 2020

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Cheol-Hyun Cho and Yong-Geun Oh, Floer cohomology and disc instantons of L agrangian torus fibers in F ano toric manifolds. Asian J. Math. 10 (2006), no. 4, 773--814

work page 2006

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Cheol-Hyun Cho and Mainak Poddar, Holomorphic orbi-discs and L agrangian F loer cohomology of symplectic toric orbifolds. J. Differential Geom. 98 (2014), no. 1, 21--116

work page 2014

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Aliakbar Daemi and Kenji Fukaya, Atiyah-Floer conjecture: a formulation, a strategy of proof and generalizations, in Modern geometry: a celebration of the work of Simon Donaldson , 23--57, Proc. Sympos. Pure Math., 99, Amer. Math. Soc., Providence, RI

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Donaldson, Boundary value problems for Yang-Mills fields

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work page 1992

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P art I , AMS/IP Studies in Advanced Mathematics, vol

Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Lagrangian intersection F loer theory: anomaly and obstruction. P art I , AMS/IP Studies in Advanced Mathematics, vol. 46.1, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009

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I , Duke Math

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work page 2010

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Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Toric degeneration and nondisplaceable Lagrangian tori in S^2 S^2 , Int. Math. Res. Not. IMRN 2012 , no. 13, 2942--2993

work page 2012

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Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Kuranishi structures and virtual fundamental chains., Springer Monogr. Math. Springer, Singapore, [2020], ©2020. xv+638 pp

work page 2020

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Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono, Construction of Kuranishi structures on the moduli spaces of pseudo-holomorphic disks: II., Adv. Math. 442 (2024), Paper No. 109561, 63 pp

work page 2024

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Masahiro Futaki and Fumihiko Sanda, Equivariant homological mirror symmetry for C and CP^1 , J. Geom. Phys. 192 (2023), Paper No. 104929, 26 pp

work page 2023

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Kenji Fukaya, Cyclic symmetry and adic convergence in Lagrangian Floer theory., Kyoto J. Math. 50, no. 3 (2010), 521--590

work page 2010

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Kenji Fukaya, Lie groupoids, deformation of unstable curves, and construction of equivariant Kuranishi charts, Publ. Res. Inst. Math. Sci. 57 (2021), no. 3-4, 1109--1225

work page 2021

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Robbin, and Dietmar Arno Salamon, The moment-weight inequality and the Hilbert-Mumford criterion—GIT from the differential geometric viewpoint

Valentina Georgoulas, Joel W. Robbin, and Dietmar Arno Salamon, The moment-weight inequality and the Hilbert-Mumford criterion—GIT from the differential geometric viewpoint. Lecture Notes in Math., 2297 Springer, Cham, (2021), vii+190 pp

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Systems 3 (1983), no

Victor Guillemin and Shlomo Sternberg, On collective complete integrability according to the method of T himm , Ergodic Theory Dynam. Systems 3 (1983), no. 2, 219--230

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Kristen Hendricks, Robert Lipshitz and Sucharit Sarkar, A simplicial construction of G -equivariant Floer homology, Proc. Lond. Math. Soc. (3) 121 (2020), no. 6, 1798--1866

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Victoria Hoskins, Geometric invariant theory and symplectic quotients, available at https://userpage.fu-berlin.de/hoskins/GITnotes.pdf https://userpage.fu-berlin.de/hoskins/GITnotes.pdf

work page

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Yoosik Kim, Lifting holomorphic disks from flag varieties to basic affine spaces, in preparation

work page

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Fixed Point Theory Appl

Yoosik Kim, Disk potential functions for quadrics., J. Fixed Point Theory Appl. 25 (2023), No. 2, Paper No. 46, 31 pp

work page 2023

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(MN-31), Volume 31

Frances Clare Kirwan, Cohomology of Quotients in Symplectic and Algebraic Geometry. (MN-31), Volume 31. Princeton University Press, 1984

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Symplectic Geom

Yoosik Kim, Siu-Cheong Lau, and Xiao Zheng, Disk potential functions for polygon spaces., J. Symplectic Geom. 22 (2024), No. 4, 735--815

work page 2024

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G. R. Kempf and L. Ness, The length of vectors in representation spaces, in Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978) , pp. 233--243, Lecture Notes in Math., 732, Springer, Berlin

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Siu-Cheong Lau, Nai-Chung Conan Leung, and Yan-Lung Leon Li, Equivariant Lagrangian correspondence and a conjecture of Teleman, (2023), arXiv:2312.13926 https://arxiv.org/abs/2312.13926

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