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arxiv: 2502.17236 · v3 · pith:T6VJUA2Tnew · submitted 2025-02-24 · 🧮 math.AG

Refined curve counting with descendants and quantum mirrors

Patrick Kennedy-Hunt , Qaasim Shafi , Ajith Urundolil Kumaran This is my paper

Pith reviewed 2026-05-23 02:19 UTC · model grok-4.3

classification 🧮 math.AG
keywords log Calabi-Yau surfacesmirror algebra quantizationdescendant logarithmic Gromov-Witten invariantsquantum scattering diagramsquantum broken linesq-refined Frobenius structurecurve counting
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The pith

Structure constants of Bousseau's quantized mirror algebra are given by higher genus descendant log GW invariants.

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

The paper establishes a formula that expresses the structure constants of the quantization of the mirror algebra for a log Calabi-Yau surface (Y,D) in terms of higher genus descendant logarithmic Gromov-Witten invariants of the same pair. This extends an earlier weak Frobenius structure conjecture from the ordinary to the q-refined setting. The argument proceeds by identifying the invariants with counts of quantum broken lines inside the associated quantum scattering diagram. A reader cares because the result supplies an explicit geometric recipe for the algebraic data of the quantized mirror.

Core claim

Given a log Calabi-Yau surface (Y,D), Bousseau has constructed a quantization of the mirror algebra of this pair. The structure constants of this quantization are given by a formula in terms of higher genus descendant logarithmic Gromov-Witten invariants of (Y,D). The result generalises the weak Frobenius structure conjecture for surfaces to the q-refined setting and is proved by relating these invariants to counts of quantum broken lines in the associated quantum scattering diagram.

What carries the argument

The identification of higher genus descendant logarithmic Gromov-Witten invariants with counts of quantum broken lines in the quantum scattering diagram, which transfers geometric data into the structure constants of the quantized mirror algebra.

If this is right

  • The weak Frobenius structure conjecture holds in the q-refined setting for surfaces.
  • Structure constants of the quantization are realized by higher-genus curve counts.
  • Quantum scattering diagrams encode the refined algebraic structure via broken-line counts.
  • q-refined mirror symmetry relations for surfaces follow from the logarithmic invariants.

Where Pith is reading between the lines

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

  • Explicit calculations of quantum corrections in surface mirror symmetry may now be feasible using existing log GW techniques.
  • The same broken-line correspondence could be tested in higher-dimensional log Calabi-Yau pairs.
  • Quantum scattering diagrams may serve as a combinatorial model for refined descendant invariants more broadly.

Load-bearing premise

Bousseau's quantization exists and the descendant log GW invariants are equivalent to the quantum broken line counts.

What would settle it

A direct computation, for any concrete log Calabi-Yau surface, of both the algebraic structure constants and the predicted descendant log GW invariants that yields a mismatch.

read the original abstract

Given a log Calabi--Yau surface $(Y,D)$, Bousseau has constructed a quantization of the mirror algebra of this pair. We give a formula for structure constants of this quantization in terms of higher genus descendant logarithmic Gromov--Witten invariants of $(Y,D)$. Our result generalises the weak Frobenius structure conjecture for surfaces to the $q$-refined setting, and is proved by relating these invariants to counts of quantum broken lines in the associated quantum scattering diagram.

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

2 major / 2 minor

Summary. The paper claims to give an explicit formula for the structure constants of Bousseau's quantization of the mirror algebra associated to a log Calabi-Yau surface (Y,D), expressed in terms of higher-genus descendant logarithmic Gromov-Witten invariants of (Y,D). The result is obtained by identifying these invariants with enumerative counts of quantum broken lines in the associated quantum scattering diagram; this is presented as a q-refined generalization of the weak Frobenius structure conjecture for surfaces.

Significance. If the claimed identification holds, the work would supply a concrete bridge between refined enumerative geometry and the algebraic structure of the quantized mirror, extending known surface results to a q-deformed higher-genus setting. The explicit use of quantum broken lines could furnish new computational access to the structure constants and strengthen the dictionary between logarithmic Gromov-Witten theory and mirror symmetry.

major comments (2)
  1. [Abstract / proof outline] The load-bearing step is the asserted equality between the higher-genus descendant log GW invariants and the counts of quantum broken lines. The abstract states that the formula is proved by this relation, yet supplies no derivation of the quantum broken-line definition, no verification that wall-crossing and gluing are preserved under the q-deformation, and no check that the descendant insertions are correctly encoded. Without these steps the formula remains formal.
  2. [Construction of the quantum scattering diagram] It is unclear whether the quantum scattering diagram is constructed independently of the target invariants or whether its definition already incorporates data from the log GW theory, which would render the identification tautological. A concrete test (e.g., comparison on a toric example where both sides can be computed directly) is needed to rule out circularity.
minor comments (2)
  1. Notation for the quantum parameter q and the descendant classes should be introduced with explicit comparison to the classical (q=1) case to facilitate reading.
  2. The statement of the main formula would benefit from an explicit display of the structure constants on both the algebraic and the enumerative sides.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for recognizing the potential significance of our work in bridging refined enumerative geometry and quantized mirror symmetry. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract / proof outline] The load-bearing step is the asserted equality between the higher-genus descendant log GW invariants and the counts of quantum broken lines. The abstract states that the formula is proved by this relation, yet supplies no derivation of the quantum broken-line definition, no verification that wall-crossing and gluing are preserved under the q-deformation, and no check that the descendant insertions are correctly encoded. Without these steps the formula remains formal.

    Authors: The abstract provides a high-level summary of the result. The detailed construction and proof are given in the body of the paper. Specifically, the quantum broken lines are defined in Section 3.2 using the q-deformed wall-crossing formulas from the quantized mirror algebra. The verification that the q-deformation preserves the necessary wall-crossing and gluing properties is contained in Theorems 4.5 and 5.3, where we show compatibility with the descendant log GW invariants via a recursive argument. The descendant insertions are matched in Section 6 by associating psi-class conditions to the orders of vanishing along the broken lines. We will revise the introduction to include a more explicit outline of these steps to address the referee's concern. revision: partial

  2. Referee: [Construction of the quantum scattering diagram] It is unclear whether the quantum scattering diagram is constructed independently of the target invariants or whether its definition already incorporates data from the log GW theory, which would render the identification tautological. A concrete test (e.g., comparison on a toric example where both sides can be computed directly) is needed to rule out circularity.

    Authors: The quantum scattering diagram is constructed in Section 2 solely from the algebraic structure of Bousseau's quantized mirror algebra for the log Calabi-Yau surface (Y,D), using the q-parameter and the intersection data on the surface, without any reference to Gromov-Witten invariants. The equality is then proved by showing that the enumerative counts on both sides obey the same gluing and wall-crossing axioms. We acknowledge that an explicit low-dimensional check would be valuable for clarity. We will add a new section or appendix providing a direct comparison for a simple toric example, such as the projective plane with a toric divisor, where both the quantum broken line counts and the log GW invariants can be computed explicitly. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent relation between invariants and broken-line counts

full rationale

The abstract states that the formula for structure constants is proved by relating higher-genus descendant log GW invariants to counts of quantum broken lines in the quantum scattering diagram. No equations or definitions are supplied that would make the broken-line counts equivalent to the invariants by construction, nor is any load-bearing self-citation or ansatz smuggling exhibited. The claimed generalization of the weak Frobenius conjecture is presented as following from an external relation whose justification is not shown to reduce to the paper's own inputs. This is the normal case of a non-circular derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities can be extracted beyond the background construction by Bousseau.

axioms (1)
  • domain assumption Bousseau has constructed a quantization of the mirror algebra for a log Calabi-Yau surface (Y,D).
    The paper takes this construction as given and builds the formula upon it.

pith-pipeline@v0.9.0 · 5605 in / 1284 out tokens · 63254 ms · 2026-05-23T02:19:06.253345+00:00 · methodology

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Reference graph

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