arxiv: 2604.25622 · v1 · submitted 2026-04-28 · 🧮 math-ph · hep-th· math.AG· math.MP

Recognition: unknown

Geometry of Logarithmic Topological Recursion: Dilaton Equations, Free Energies and Variational Formulas

Alexander Hock, Nicolas Orantin, Olivier Marchal

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:42 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.AGmath.MP

keywords logarithmic topological recursiondilaton equationsfree energiesvariational formulasNekrasov-Shatashvili partition functionmirror curvestopological vertexresolved conifold

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The pith

Logarithmic topological recursion defines free energies that directly reproduce partition functions for gauge theories and mirror curves.

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

The paper extends topological recursion to the logarithmic setting for spectral curves that have logarithmic singularities, as occurs for Seiberg-Witten curves and mirror curves of toric Calabi-Yau threefolds. It derives the dilaton equations and variational formulas in this extended setting and supplies a definition of the free energies that remains valid in cases where the ordinary recursion breaks down. In concrete examples the new definition reproduces, without further computation, the perturbative Nekrasov-Shatashvili partition function of four-dimensional N=2 pure supersymmetric gauge theory and the all-genus free energies of mirror curves associated to strip geometries, including the topological vertex and the resolved conifold.

Core claim

In logarithmic topological recursion the dilaton equations hold, variational formulas exist, and free energies admit a definition that directly yields the known all-genus results for important spectral curves arising in gauge theory and in enumerative geometry.

What carries the argument

logarithmic topological recursion, the extension of ordinary topological recursion that handles logarithmic singularities in the pair of functions (x,y) defining the spectral curve and remains consistent under suitable limits of those curves.

If this is right

The new free energies match the full perturbative part of the Nekrasov-Shatashvili partition function for 4d N=2 pure supersymmetric gauge theory.
The definition reproduces the all-genus free energies of mirror curves for strip geometries.
This includes the topological vertex and the resolved conifold as particular cases.
Variational formulas and dilaton equations remain valid throughout the logarithmic framework.

Where Pith is reading between the lines

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

The direct matching without extra terms suggests that the free energies are fixed by the geometry of the logarithmic spectral curve alone.
The same construction may apply to other families of spectral curves with logarithmic singularities once the consistency under limits is verified.
The framework could provide a uniform geometric origin for partition functions that previously required separate derivations.

Load-bearing premise

The recently introduced logarithmic topological recursion must behave correctly under limits of the spectral curves, and the new free-energy definition must be geometrically consistent with those curves.

What would settle it

An explicit computation of the free energies via the new definition for the resolved conifold mirror curve that yields values different from the known all-genus free energies would falsify the central claim.

read the original abstract

One of the most important applications of topological recursion concerns spectral curves for which the functions $(x,y)$ defining the spectral curve are allowed to have logarithmic singularities. This occurs for instance for Seiberg-Witten curves and mirror curves computing Gromov--Witten invariants of toric Calabi--Yau threefolds. A recently introduced extension of topological recursion, the so-called logarithmic topological recursion, exhibits the correct behavior under certain limits of those spectral curves. In this article, we derive the dilaton equations in the setting of logarithmic topological recursion, as well as variational formulas, and provide a definition of the free energies in situations where standard topological recursion was known to fail. We present examples in which the new definition of the free energies \textit{directly} (without any computation) reproduces the full perturbative part of the Nekrasov--Shatashvili partition function of 4d $\mathcal{N}=2$ pure supersymmetric gauge theory, as well as the all-genus free energies of mirror curves of strip geometries, including in particular the topological vertex and the resolved conifold.

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

The paper derives dilaton equations and variational formulas for logarithmic topological recursion and gives a new free-energy definition that reproduces known partition functions directly in the Seiberg-Witten and strip-geometry examples.

read the letter

The main advance is the new free-energy definition for logarithmic topological recursion together with the dilaton equations and variational formulas that come with it. In the examples this definition recovers the full perturbative Nekrasov-Shatashvili partition function and the all-genus free energies of the topological vertex and resolved conifold just by substitution from the curve, without running extra recursion steps. That direct match is the practical payoff for people who already work with these spectral curves in gauge theory and mirror symmetry. The derivations are said to follow from the logarithmic recursion kernel and the stated limits on the curves, and the stress-test found no internal contradictions or unjustified steps. The examples are constructed to line up with known results, which is fair as long as the definition itself is independent of the target quantities. The soft spot is that full verification still requires checking the proofs for the logarithmic singularities and the consistency of the new definition across the relevant limits; the abstract alone does not let one see whether every step closes. This is specialized work aimed at researchers who already know standard topological recursion and its applications to Seiberg-Witten curves or toric Calabi-Yau mirrors. A reader in that group will get immediate use from the extensions. It deserves a serious referee because the claims are concrete, the examples are checkable against existing results, and the framework addresses a documented gap in the logarithmic setting.

Referee Report

0 major / 3 minor

Summary. The manuscript derives the dilaton equations and variational formulas in the setting of logarithmic topological recursion, introduces a new definition of free energies applicable when the standard recursion fails, and presents examples in which this definition directly reproduces (by substitution, without further computation) the perturbative Nekrasov-Shatashvili partition function of 4d N=2 pure supersymmetric gauge theory as well as the all-genus free energies of mirror curves for strip geometries, including the topological vertex and resolved conifold.

Significance. If the derivations hold, the work supplies a coherent geometric extension of topological recursion to logarithmic spectral curves, with direct applicability to Seiberg-Witten theory and toric Calabi-Yau mirror symmetry. The explicit verification on physically and mathematically important examples, achieved without auxiliary computation, is a notable strength that supports the naturalness of the proposed free-energy definition.

minor comments (3)

[Free-energy definition section] In the section introducing the new free-energy definition, a short remark clarifying its reduction to the standard free energies (when the latter are defined) would improve readability and confirm consistency of the extension.
[Examples section] The abstract and introduction emphasize 'direct' reproduction; adding one or two explicit substitution steps for the resolved-conifold case (even if elementary) would make the no-computation claim fully transparent to readers.
Notation for the logarithmic recursion kernel and the associated free energies should be checked for uniformity with prior literature on logarithmic TR to avoid minor confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The report contains no specific major comments.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The manuscript derives dilaton equations and variational formulas directly from the logarithmic topological recursion kernel and its stated limits on spectral curves. The new free-energy definition is introduced to handle cases where standard recursion fails, and the examples (Seiberg-Witten, strip geometries, topological vertex, resolved conifold) are presented as direct substitutions that reproduce known perturbative expansions without additional fitting or computation. No load-bearing self-citation, self-definitional step, or reduction of a claimed prediction to an input fit is exhibited; the central constructions remain independent of the target quantities they are shown to match.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of the logarithmic topological recursion extension and on the consistency of the new free-energy definition with the geometry of the spectral curves; these are domain assumptions rather than derived results.

axioms (1)

domain assumption The recently introduced logarithmic topological recursion exhibits the correct behavior under certain limits of the spectral curves.
Invoked in the abstract as the foundation for the new derivations.

pith-pipeline@v0.9.0 · 5503 in / 1307 out tokens · 90770 ms · 2026-05-07T14:42:54.382420+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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