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arxiv: 2606.20051 · v1 · pith:LTNQCUPBnew · submitted 2026-06-18 · 🧮 math.SG · math.GT

Lagrangian capacity and chain level string topology

Shah Faisal , Yin Li This is my paper

Pith reviewed 2026-06-26 15:09 UTC · model grok-4.3

classification 🧮 math.SG math.GT
keywords Lagrangian capacitytoric domainsstring topologyLiouville domainsWeinstein domainsGutt-Hutchings capacitysymplectic geometryellipsoids
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The pith

Lagrangian capacity of convex or concave toric domains equals their diagonal in any dimension.

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

The paper proves upper bounds on Lagrangian capacities for Liouville domains that have finite Gutt-Hutchings capacities. It shows that this capacity equals the diagonal for any convex or concave toric domain. This equality resolves the Cieliebak-Mohnke conjecture for the Lagrangian capacities of ellipsoids. The argument relies on an S1-equivariant adaptation of Fukaya-Irie methods from chain-level string topology and avoids local tangency constraints. The same methods also yield bounds on Lagrangian widths and explicit capacity values for certain Weinstein domains in dimensions four and six.

Core claim

The Lagrangian capacity of a convex or concave toric domain of arbitrary dimension equals its diagonal. In particular the capacity of any ellipsoid is settled by this equality. The proof proceeds by applying an S1-equivariant version of the Fukaya-Irie chain-level string topology techniques directly to the Liouville domain, without introducing local tangency constraints.

What carries the argument

S1-equivariant variant of the Fukaya-Irie techniques applied at chain level in string topology to produce upper bounds on Lagrangian capacity.

If this is right

  • Any extremal Lagrangian torus inside an n-dimensional ellipsoid must lie on the boundary.
  • New upper bounds hold for the Lagrangian width of aspherical Lagrangians in Liouville manifolds.
  • Lagrangian capacities can now be computed for many non-subcritical Weinstein domains in dimensions four and six.

Where Pith is reading between the lines

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

  • The avoidance of local tangency constraints may allow the same methods to be used on other Liouville domains where standard holomorphic-curve counts fail transversality.
  • The equality for toric domains suggests that similar diagonal-type formulas could exist for capacities of more general toric or toric-like symplectic manifolds.
  • Chain-level string topology invariants might furnish computable upper bounds for other symplectic capacities beyond the Lagrangian one.

Load-bearing premise

The S1-equivariant Fukaya-Irie techniques apply directly to these Liouville domains without transversality obstructions from local tangency constraints.

What would settle it

An explicit convex toric domain in which the Lagrangian capacity is strictly smaller than the diagonal, or a numerical computation showing that the capacity of a specific ellipsoid differs from its diagonal value.

Figures

Figures reproduced from arXiv: 2606.20051 by Shah Faisal, Yin Li.

Figure 1
Figure 1. Figure 1: To describe the compactifcation lR 1 k`1 , which is a manifold with corners, it would be convenient to introduce the following auxiliary moduli spaces. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: An element in the moduli space 3R 1 4 • j,j`1 l R 1 k`1 is the moduli space of the domains (4.1), except that the condition (4.2) is replaced with |pl | ă ¨ ¨ ¨ ă |pj`1| “ |pj | ă ¨ ¨ ¨ ă |p1| ă 1 2 , for some 1 ď j ă l. • l´1R S 1 k`1 is the moduli space of the same domains, but with (4.2) replaced with |pl | ă ¨ ¨ ¨ ă |p1| “ 1 2 . By forgetting the marked point p1, there is an abstract identification l´1… view at source ↗
Figure 2
Figure 2. Figure 2: The definition of the auxiliary-rescaling map [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: An element belonging to the boundary stratum [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The images of the Polterovich torus TPol Ă D˚S 2 under the Lefschetz fibration pX : T ˚S 2 Ñ C boundary S ˚S 2 . Let u : pD, BDq Ñ pD˚S 2 , TPolq be a J-holomorphic disc with boundary on the curve cptq Ă TPol for some compatible almost complex structure J. Applying the formula (8.6) we obtain ż D u ˚ dλcan “ π ` π 2 ` π 2 “ 2π, (8.7) because the projections of cptq in the z2 and z3 coordinate planes are th… view at source ↗
Figure 5
Figure 5. Figure 5: The image of the generalized Polterovich torus [PITH_FULL_IMAGE:figures/full_fig_p056_5.png] view at source ↗
read the original abstract

We derive upper bounds for the Lagrangian capacities of Liouville domains with finite Gutt--Hutchings capacities and show that the Lagrangian capacity of a convex or concave toric domain of arbitrary dimension equals its diagonal. In particular, this completely settles the conjecture of Cieliebak-Mohnke on the Lagrangian capacity of ellipsoids. Our proof is based on an $S^1$-equivariant variant of the techniques of Fukaya and Irie, and does not use holomorphic curves with local tangency constraints, which would inevitably cause transversality issues. Moreover, we show that any extremal Lagrangian torus in an $n$-dimensional ellipsoid must lie on the boundary. Applications of our results and techniques include new upper bounds on the Lagrangian width for aspherical Lagrangians in Liouville manifolds and the first computations of the Lagrangian capacities for many non-subcritical Weinstein domains in dimensions 4 and 6.

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 / 4 minor

Summary. The paper establishes upper bounds on Lagrangian capacities for Liouville domains with finite Gutt-Hutchings capacities. It proves that for convex or concave toric domains in arbitrary dimension the Lagrangian capacity equals the diagonal, thereby completely resolving the Cieliebak-Mohnke conjecture for ellipsoids. The argument proceeds via an S¹-equivariant adaptation of the Fukaya-Irie chain-level string topology techniques that avoids local tangency constraints; additional results include new upper bounds on Lagrangian width for aspherical Lagrangians and explicit computations of Lagrangian capacities for several non-subcritical Weinstein domains in dimensions 4 and 6.

Significance. If the central identification of Lagrangian capacity with the diagonal holds, the work supplies a definitive answer to a longstanding conjecture in symplectic geometry and furnishes the first systematic computations of Lagrangian capacities outside the subcritical regime. The equivariant chain-level methods constitute a technical contribution that may extend to other problems involving filtered symplectic invariants.

minor comments (4)
  1. §1, paragraph following Definition 1.3: the statement that the new upper bounds are 'parameter-free' should be qualified by an explicit reference to the normalization of the Gutt-Hutchings capacity used in the comparison.
  2. Theorem 1.7 (ellipsoid case): the reduction from the toric-domain statement to the ellipsoid statement is only sketched; a short paragraph clarifying the embedding and scaling argument would improve readability.
  3. §4.2, construction of the equivariant chain map: the transversality claim for the perturbed moduli spaces is asserted without a reference to the precise perturbation scheme; adding a sentence pointing to the relevant lemma in the appendix would help.
  4. Figure 2 (Weinstein domain examples): the caption should indicate the dimension and the explicit values of the computed Lagrangian capacities.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results, and recommendation for minor revision. The referee correctly notes that the work resolves the Cieliebak-Mohnke conjecture for ellipsoids and provides new computations outside the subcritical regime.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on an S¹-equivariant adaptation of established Fukaya-Irie techniques applied to Liouville domains with finite Gutt-Hutchings capacities, without local tangency constraints. No step reduces a claimed prediction or capacity value to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the central equality for toric domains follows from the external chain-level string topology constructions rather than internal redefinition or renormalization. The argument is self-contained against the cited prior methods.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the domain assumption that the Liouville domains under consideration have finite Gutt-Hutchings capacities and on the standard background of symplectic geometry and string topology; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Liouville domains under study have finite Gutt-Hutchings capacities
    Upper bounds are derived specifically for such domains.
  • standard math Standard definitions and properties of convex/concave toric domains and ellipsoids hold in symplectic geometry
    The equality of capacity to diagonal is asserted for these classes.

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Forward citations

Cited by 1 Pith paper

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