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arxiv: 2604.14077 · v1 · submitted 2026-04-15 · 🧮 math-ph · math.MP· math.RT· nlin.SI

Open WDVV equations and bigvee-systems

Alessandro Proserpio , Ian A. B. Strachan This is my paper

Pith reviewed 2026-05-10 11:53 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.RTnlin.SI
keywords open WDVV equationsV-systemsrank-one extensionsrational solutionscovectorsGromov-Witten theorysuperpotentialsalmost-duality
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The pith

Rank-one extensions of V-systems yield rational solutions to open WDVV equations when covectors meet specific algebraic and geometric conditions.

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

This paper generalizes Veselov's V-systems, which give algebraic and geometric conditions for rational solutions to the WDVV associativity equations, to the open WDVV equations coming from open Gromov-Witten theory. It focuses on rank-one extensions and derives additional conditions on the covectors that, when added to the V-system, ensure the resulting system solves the open equations rationally. These conditions are presented along with examples that relate the solutions to superpotentials and Dubrovin's almost-duality. A reader would care because this provides a concrete way to build rational solutions in the open setting, extending the reach of algebraic methods in integrable systems and enumerative geometry.

Core claim

For rank-one extensions of a V-system, algebraic and geometric conditions on the supplementary covectors allow the construction of rational solutions to the open WDVV equations.

What carries the argument

The V-system supplemented by rank-one extension conditions on covectors, which enforce the rationality of solutions to the open WDVV equations.

Load-bearing premise

Open WDVV equations admit rational solutions precisely when the covectors satisfy the supplementary algebraic and geometric conditions for rank-one extensions of a V-system.

What would settle it

Finding a rank-one extension of a V-system whose covectors do not satisfy the new conditions but still give a rational solution to the open WDVV equations, or one that satisfies the conditions but fails to solve the equations.

Figures

Figures reproduced from arXiv: 2604.14077 by Alessandro Proserpio, Ian A. B. Strachan.

Figure 1
Figure 1. Figure 1: Different differences (β○ − β1 ≠ β○ − β2) defining the same hyperplane Hα The left-hand-side requires more care as different differences could define the same hyperplane (see Figure [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geometry of non-crystallographic root systems The Coxeter group G2. This example will be included in the next section as a special case of a dihedral group. Since G2 has long and short roots one has different constants hl and hs (as in the Bn-case). The construction places restriction on these otherwise free data. 4.2. Non-crystallographic examples The crystallographic examples above rested on the classifi… view at source ↗
read the original abstract

The idea of a $\bigvee$-system was introduced by Veselov in the study of rational solutions of the WDVV equations of associativity. These are algebraic/geometric conditions on the set of covectors that appear in rational solutions to the WDVV equations. Here, this idea is generalized to open WDVV equations, which are an additional set of PDEs originating from open Gromow-Witten Theory. We develop -- for rank-one extensions -- algebraic/geometric conditions on the covectors that supplement the $\bigvee$-system to give rational solutions to the open WDVV equations. Examples, and the relation to superpotentials and to Dubrovin almost-duality, are given.

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

1 major / 3 minor

Summary. The paper generalizes Veselov's ∨-systems—algebraic/geometric conditions on covectors yielding rational solutions to the WDVV equations—to the open WDVV equations from open Gromov-Witten theory. For rank-one extensions of a ∨-system, it derives supplementary algebraic and geometric conditions on the covectors that ensure rational solutions to the full open system. Examples are constructed, and relations to superpotentials and Dubrovin almost-duality are discussed.

Significance. If the supplementary conditions are correctly derived and verified, the work supplies a constructive method for producing rational solutions in the open case, extending the closed WDVV theory in a controlled way. The explicit examples and the links to almost-duality and superpotentials add practical value and may facilitate further study of open topological field theories and related integrable systems.

major comments (1)
  1. [§3] §3, main theorem on supplementary conditions: the sufficiency argument proceeds by direct substitution into the open WDVV PDEs, but the separation between the inherited ∨-system associativity conditions and the new open-sector conditions is not made fully explicit; a short auxiliary lemma isolating the open PDEs would strengthen the claim that the extension preserves rationality without hidden dependencies.
minor comments (3)
  1. [Introduction] The introduction would benefit from a one-sentence reminder of the precise form of the open WDVV equations (as opposed to the closed ones) to orient readers unfamiliar with the open Gromov-Witten origin.
  2. [Examples] In the examples section, the explicit rational solutions are stated but the verification that they satisfy the open equations is only sketched; adding a short table or inline check for the lowest-rank case would improve readability.
  3. [§2] Notation for the covectors and the rank-one extension parameter is consistent within sections but could be collected in a short notation table at the end of §2 for quick reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestion for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: §3, main theorem on supplementary conditions: the sufficiency argument proceeds by direct substitution into the open WDVV PDEs, but the separation between the inherited ∨-system associativity conditions and the new open-sector conditions is not made fully explicit; a short auxiliary lemma isolating the open PDEs would strengthen the claim that the extension preserves rationality without hidden dependencies.

    Authors: We agree that the separation can be made more explicit for clarity. In the revised manuscript we will insert a short auxiliary lemma immediately preceding the main theorem. The lemma will state the open WDVV equations in isolation, assume the closed ∨-system associativity conditions, and derive that the remaining open-sector PDEs reduce exactly to the stated supplementary algebraic and geometric conditions on the covectors. This will confirm that rationality is preserved without hidden dependencies. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper generalizes Veselov's ∨-system (external citation) by constructing supplementary algebraic/geometric conditions on covectors for rank-one extensions that produce rational solutions to the open WDVV equations. These conditions are developed as new supplements rather than being defined in terms of the target solutions or fitted to them. No self-definitional loops, predictions that reduce to input fits, load-bearing self-citations, or ansatz smuggling appear in the stated approach, examples, or relations to superpotentials/Dubrovin duality. The central claim remains independent and constructive.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated. The work builds on Veselov's ∨-systems and the definition of open WDVV from open Gromov-Witten theory without introducing new postulated objects in the summary provided.

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