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arxiv: 2512.24382 · v2 · submitted 2025-12-30 · 🧮 math.SG

Recognition: 2 theorem links

· Lean Theorem

Equivariant Partially Wrapped Fukaya Categories on Liouville Sectors

Dongwook Choa , Jiawei Hu , Siu-Cheong Lau , Yan-Lung Leon Li
Authors on Pith no claims yet

Pith reviewed 2026-05-16 18:54 UTC · model grok-4.3

classification 🧮 math.SG
keywords equivariant Floer theoryLiouville sectorssymplectic quotientnodal singularitiesLagrangian Floer cohomologyFukaya categoriesLekili-Segal conjecturewrapped Fukaya categories
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The pith

An equivariant correspondence establishes an isomorphism between Lagrangian Floer cohomologies of symmetric Liouville manifolds and their nodal symplectic quotients.

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

The authors construct an equivariant Lagrangian Floer theory for Liouville sectors equipped with a Lie group symmetry. They introduce a correspondence that connects the equivariant cohomology computed in the symmetric space to the cohomology of the quotient space. In cases where the quotient develops nodal singularities, this correspondence becomes an isomorphism, thereby confirming a long-standing conjecture by Lekili and Segal.

Core claim

We develop an equivariant Lagrangian Floer theory for Liouville sectors with G-symmetry. For Liouville manifolds with G-symmetry, we develop a correspondence theory to relate the equivariant Lagrangian Floer cohomology upstairs and Lagrangian Floer cohomology of its quotient. Furthermore, we study the symplectic quotient in the presence of nodal type singularities and prove that the equivariant correspondence gives an isomorphism on cohomologies which was conjectured by Lekili-Segal.

What carries the argument

The equivariant correspondence theory relating partially wrapped Fukaya categories on the G-symmetric Liouville sector to the Lagrangian Floer cohomology of its symplectic quotient.

If this is right

  • The Lekili-Segal conjecture holds for symplectic quotients with nodal singularities.
  • Equivariant Floer cohomology can be used to compute the cohomology of the quotient manifold.
  • Partially wrapped Fukaya categories extend naturally to the equivariant setting under group actions.
  • Invariants of singular symplectic manifolds become accessible through the upstairs equivariant data.

Where Pith is reading between the lines

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

  • The correspondence might extend to other singularity types if the nodal condition can be weakened with additional structure.
  • This could simplify computations in homological mirror symmetry for orbifold or quotient geometries.
  • Explicit low-dimensional examples with known group actions would provide direct tests of the isomorphism.

Load-bearing premise

The Liouville sectors must admit a G-symmetry and the resulting symplectic quotient must have only nodal-type singularities.

What would settle it

An explicit computation of both the equivariant cohomology and the quotient cohomology on a concrete example with nodal singularities that shows the two sides are not isomorphic.

Figures

Figures reproduced from arXiv: 2512.24382 by Dongwook Choa, Jiawei Hu, Siu-Cheong Lau, Yan-Lung Leon Li.

Figure 1
Figure 1. Figure 1: Configurations of pearl trajectories that define the maps PSSN and PSS−1 N . Up to generic choices, P  a, x; H(N), fbN  is a smooth manifold. By our choice of Hamiltonians, we have E(u) < ∞, and hence Gromov-Floer compactness applies. Consequently, we can compactify the one-dimensional stratum of P [PITH_FULL_IMAGE:figures/full_fig_p042_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Quasi-units are natural with respect to continuation maps and approximations. Proposition 5.14. The horizontal squares and triangles in Diagram 2 commute up to homotopy, while the vertical squares commute strictly. Proof. The homotopy commutativity of the horizontal squares and triangles follows from an argument identitical to that in [AA24, Lemma 3.19]. The strict commutativity of the vertical squares is … view at source ↗
Figure 3
Figure 3. Figure 3: Quilted Strips Defining the Bimodule. The quilted maps u = (u1, u2) : Σ = Σ1 ∪ Σ2 → X//G × XG with Lagrangian boundary conditions (L ′ 0 , · · ·L ′ k ′) ⊆ X//G, (L0,G, · · · , Lk,G) ⊆ XG and seam condition L π G, are described as follows [PITH_FULL_IMAGE:figures/full_fig_p059_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Composition of [µ 0|1|1 (1L, −)] and its left inverse. Indeed, when both input and output are the element 1L, the only quilted configurations that can contribute to the strip component in [PITH_FULL_IMAGE:figures/full_fig_p063_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: CF• S 1 (L, L; Kν) for ν = 0 and ν = 1. This computation generalizes to Lagrangians in a conic fibration. Definition 7.2. Let X be a conic fibration, and let L ⊂ µ −1 (0) be an exact G-invariant Lagrangian. We say that L is of disk type near F if, in the local model (7.2) near F, it can be described as   Dℜ L ∩ N L ′   ⊂   D 2 N ≃ DNF/X F   , (7.9) i.e., a Lagrangian disk bundle over a … view at source ↗
read the original abstract

We develop an equivariant Lagrangian Floer theory for Liouville sectors that have symmetry of a Lie group $G$. Moreover, for Liouville manifolds with $G$-symmetry, we develop a correspondence theory to relate the equivariant Lagrangian Floer cohomology upstairs and Lagrangian Floer cohomology of its quotient. Furthermore, we study the symplectic quotient in the presence of nodal type singularities and prove that the equivariant correspondence gives an isomorphism on cohomologies which was conjectured by Lekili-Segal.

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.

Referee Report

0 major / 3 minor

Summary. The paper develops an equivariant Lagrangian Floer theory for Liouville sectors equipped with a Lie group G-action. It constructs a correspondence relating the equivariant partially wrapped Fukaya category (or its cohomology) of a G-symmetric Liouville manifold to the ordinary Lagrangian Floer cohomology of the symplectic quotient. For quotients exhibiting nodal-type singularities, the correspondence is shown to induce an isomorphism on cohomology, confirming a conjecture of Lekili-Segal.

Significance. If the technical constructions hold, the work supplies a systematic equivariant extension of partially wrapped Fukaya categories and a quotient correspondence that directly resolves the Lekili-Segal conjecture in the nodal case. This furnishes a new tool for computing symplectic invariants under group actions and may connect to mirror symmetry statements involving orbifolds or singular quotients.

minor comments (3)
  1. §1.2: the statement of the main isomorphism theorem should explicitly reference the precise nodal singularity hypothesis (e.g., the local model for the quotient) rather than leaving it implicit in the correspondence construction.
  2. Definition 3.4 and §4.1: clarify whether the G-equivariant transversality is achieved by a perturbation scheme that preserves the Liouville structure or requires an auxiliary almost-complex structure; the current wording leaves the compactness argument for the moduli spaces slightly underspecified.
  3. Theorem 5.3: the proof sketch of the isomorphism on cohomology would benefit from an explicit diagram chase showing how the nodal singularity is absorbed into the correspondence functor; a short commutative diagram would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on equivariant Lagrangian Floer theory for Liouville sectors with G-action, the quotient correspondence, and the proof of the Lekili-Segal conjecture in the nodal case. We appreciate the recommendation for minor revision and the assessment that the constructions furnish a new tool for symplectic invariants under group actions.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs an equivariant Lagrangian Floer theory for G-symmetric Liouville sectors, develops a correspondence relating equivariant cohomology upstairs to the quotient's Floer cohomology, and applies this to prove an isomorphism in the nodal singularity case that confirms the external Lekili-Segal conjecture. No quoted equations or steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central claims rest on new extensions of standard Floer constructions rather than renaming or rederiving their own premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard assumptions from symplectic geometry about Liouville sectors and Floer theory; no free parameters, new entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • domain assumption Liouville sectors with G-symmetry admit well-defined equivariant Lagrangian Floer theory and quotient constructions.
    Invoked to develop the correspondence and prove the isomorphism.

pith-pipeline@v0.9.0 · 5383 in / 1163 out tokens · 32335 ms · 2026-05-16T18:54:28.434005+00:00 · methodology

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 3 internal anchors

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