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arxiv: 2406.17852 · v2 · submitted 2024-06-25 · ✦ hep-th · math-ph· math.AG· math.MP

Non-perturbative topological strings from resurgence

Murad Alim This is my paper

Pith reviewed 2026-05-24 00:14 UTC · model grok-4.3

classification ✦ hep-th math-phmath.AGmath.MP
keywords topological stringsCalabi-Yau threefoldsresurgenceBorel summationGopakumar-Vafa invariantssheaf invariantsnon-perturbative partition functionresolved conifold
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The pith

The topological string partition function on any Calabi-Yau threefold factors into resolved-conifold partition functions raised to powers of sheaf invariants.

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

The paper proves that in the holomorphic limit, the perturbative partition function of topological strings on an arbitrary Calabi-Yau threefold factors into a product where each term is the resolved conifold partition function evaluated at shifted arguments and raised to the power of specific sheaf invariants. This factorization is then used to define a non-perturbative completion by replacing each factor with its Borel sum, which is an analytic function in the string coupling. The Borel transform of the full series is given in terms of the sheaf invariants, allowing explicit computation of the Stokes jumps that provide the non-perturbative corrections. These jumps depend solely on the genus-zero Gopakumar-Vafa invariants, and their total sum is captured by a deformation of the prepotential. A reader would care because this supplies a concrete way to define the full non-perturbative topological string partition function beyond its asymptotic perturbative expansion.

Core claim

We prove that the partition function of topological strings of any CY in this limit can be written as a product, where each factor is given by the partition function of the resolved conifold with shifted arguments, raised to the power of certain sheaf invariants. We use this result to put forward an expression for the non-perturbative topological string partition function in this limit, as a product over analytic functions in the topological string coupling which correspond to the Borel sums for the resolved conifold found previously. We furthermore find an expression for the Borel transform of the full asymptotic series in this limit expressed in terms of the sheaf invariants. We use this,

What carries the argument

The exact factorization of the topological string partition function into resolved-conifold partition functions raised to sheaf-invariant powers, which carries the argument from perturbative asymptotics to non-perturbative Borel sums.

Load-bearing premise

The moduli space geometry of an arbitrary Calabi-Yau threefold permits an exact factorization of its topological string partition function into resolved-conifold factors raised to sheaf-invariant powers in the holomorphic limit.

What would settle it

An explicit computation of the partition function for a Calabi-Yau threefold whose invariants are independently known, showing that it cannot be expressed as the predicted product of resolved-conifold factors with the stated powers.

Figures

Figures reproduced from arXiv: 2406.17852 by Murad Alim.

Figure 1
Figure 1. Figure 1: Illustration of some of the singularity rays in the Borel plane [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
read the original abstract

The partition function of topological string theory on any family of Calabi-Yau threefolds is defined perturbatively as an asymptotic series in the topological string coupling and encodes, in a holomorphic limit, higher genus Gromov-Witten as well as Gopakumar-Vafa invariants. We prove that the partition function of topological strings of any CY in this limit can be written as a product, where each factor is given by the partition function of the resolved conifold with shifted arguments, raised to the power of certain sheaf invariants. We use this result to put forward an expression for the non-perturbative topological string partition function in this limit, as a product over analytic functions in the topological string coupling which correspond to the Borel sums for the resolved conifold found previously. We furthermore find an expression for the Borel transform of the full asymptotic series in this limit expressed in terms of the sheaf invariants. We use this to define the Borel sums and compute the corresponding Stokes jumps which constitute non-perturbative corrections to the partition function. The jumps depend only on genus zero GV invariants and their sum can be expressed entirely in terms of a single function which is introduced as a deformation of the prepotential.

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 prove that, in the holomorphic limit, the topological string partition function on an arbitrary Calabi-Yau threefold factors exactly as a product of resolved-conifold partition functions (with shifted arguments) each raised to the power of certain sheaf invariants. This factorization is then used to construct an explicit non-perturbative completion as a product of Borel sums, to write the Borel transform of the full asymptotic series in terms of the sheaf invariants, and to obtain Stokes jumps that depend only on genus-zero GV invariants and can be expressed via a deformation of the prepotential.

Significance. If the factorization holds, the result would supply a concrete, sheaf-invariant-based route to non-perturbative topological-string amplitudes on general CY3 from the already-known conifold Borel sums, together with an explicit Stokes-jump formula controlled by genus-zero data alone. The explicit Borel-transform expression in terms of sheaf invariants would also be a useful technical tool for resurgence calculations.

major comments (2)
  1. [§3] §3 (Factorization theorem): the central claim that the holomorphic-limit partition function factors exactly into conifold factors raised to sheaf-invariant powers, with no residual contributions from other moduli-space loci, is load-bearing for every subsequent result; the manuscript invokes the holomorphic limit but does not supply the intermediate geometric lemmas or explicit verification that the product reproduces the full set of GV/GW invariants for a general CY3 with multiple conifold points.
  2. [Eq. (5.7)] Eq. (5.7) (Stokes-jump formula): the assertion that the jumps depend only on genus-zero GV invariants is derived from the product formula, yet the cancellation of all higher-genus and non-factorizable terms is not shown explicitly; without this step the claimed independence on higher data does not follow.
minor comments (2)
  1. [Notation] The precise definition of the shifted arguments appearing in the conifold factors should be written out explicitly rather than left implicit.
  2. [§5] A short table comparing the new Stokes jumps with the known conifold case would help the reader assess the reduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comments. We address the two major points below, clarifying the structure of the arguments while committing to revisions that make the intermediate steps more explicit.

read point-by-point responses

  1. Referee: §3 (Factorization theorem): the central claim that the holomorphic-limit partition function factors exactly into conifold factors raised to sheaf-invariant powers, with no residual contributions from other moduli-space loci, is load-bearing for every subsequent result; the manuscript invokes the holomorphic limit but does not supply the intermediate geometric lemmas or explicit verification that the product reproduces the full set of GV/GW invariants for a general CY3 with multiple conifold points.

    Authors: The factorization is derived in §3 by writing the holomorphic-limit partition function in terms of its GV expansion and substituting the known product formula for each resolved-conifold factor raised to the appropriate sheaf-invariant power. Because the sheaf invariants are defined to count the contributions localized at each conifold point, the product reproduces the full set of invariants by construction once the holomorphic limit isolates those loci. We acknowledge that the geometric steps connecting the sheaf invariants to the moduli-space stratification are only sketched; in the revision we will insert a dedicated subsection with the missing lemmas and an explicit check on a threefold with several conifold points. revision: yes

  2. Referee: Eq. (5.7) (Stokes-jump formula): the assertion that the jumps depend only on genus-zero GV invariants is derived from the product formula, yet the cancellation of all higher-genus and non-factorizable terms is not shown explicitly; without this step the claimed independence on higher data does not follow.

    Authors: The jump formula follows directly from the product structure: each conifold Borel sum contributes a jump controlled by its own genus-zero data, and the overall jump is therefore assembled from genus-zero GV invariants alone. The higher-genus pieces cancel because they appear with the same coefficients on both sides of the Stokes relation for every factor. We agree that an explicit line-by-line cancellation is not written out; the revised manuscript will contain this calculation immediately before Eq. (5.7). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on stated proof of factorization

full rationale

The paper explicitly claims to prove that the partition function in the holomorphic limit factors as a product of resolved-conifold partition functions raised to sheaf-invariant powers. This factorization is then used to construct Borel sums and Stokes jumps expressed solely in terms of genus-zero GV invariants (independent data) and prior conifold Borel sums. No quoted step reduces the central claim to a fitted parameter, self-definition, or unverified self-citation chain; the derivation is presented as self-contained once the factorization proof is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; ledger entries are therefore minimal and provisional.

axioms (2)
  • domain assumption Existence of the holomorphic limit of the topological string partition function on any Calabi-Yau threefold
    Required for the product factorization to be stated.
  • domain assumption Resolved conifold partition function admits known Borel sums that can be shifted and multiplied
    Used to define the non-perturbative factors.

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