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arxiv: 2309.12046 · v2 · submitted 2023-09-21 · ✦ hep-th · math-ph· math.MP

Non-Perturbative Real Topological Strings

Marcos Mari\~no , Maximilian Schwick This is my paper

Pith reviewed 2026-05-24 06:18 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords real topological stringsresurgenceholomorphic anomaly equationsStokes constantsdisk invariantsmulti-instanton amplitudesCalabi-Yau manifoldstrans-series
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0 comments X

The pith

Extending the operator formalism produces trans-series solutions for real topological strings where disk invariants serve as Stokes constants.

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

The paper develops trans-series solutions to the holomorphic anomaly equations governing Walcher's real topological string on general Calabi-Yau manifolds. It does so by extending the operator formalism previously used for closed topological strings, yielding explicit expressions for multi-instanton amplitudes at all orders in the string coupling. The integer invariants that count disks emerge directly as the Stokes constants controlling the resurgence. A reader would care because this supplies a concrete non-perturbative completion that ties perturbative amplitudes to instanton effects through resurgence, with the disk counts playing a distinguished role.

Core claim

Trans-series solutions to the holomorphic anomaly equations for the real topological string are obtained at all orders by extending the closed-string operator formalism, explicit multi-instanton amplitudes are derived, and the integer invariants counting disks are identified as the Stokes constants in the resurgent structure.

What carries the argument

The extension of the closed topological string operator formalism to the real case, which generates the trans-series solutions and identifies disk invariants as Stokes constants.

If this is right

  • Explicit multi-instanton amplitudes become available for real topological strings on arbitrary Calabi-Yau manifolds.
  • The resurgent structure of the real topological string is completely determined by the disk-counting invariants.
  • The same Stokes constants govern the non-perturbative completion at every order in the string coupling.
  • The construction applies uniformly to general Calabi-Yau targets, not just special cases.

Where Pith is reading between the lines

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

  • The same extension technique may supply non-perturbative information for other open-string or real-string setups whose perturbative data are already known.
  • Links between real topological strings and ordinary topological strings could be sharpened by comparing their respective Stokes data.
  • The identification of disk invariants with Stokes constants offers a concrete test bed for resurgence methods in string theory beyond the closed sector.

Load-bearing premise

The operator formalism developed for closed topological strings extends directly to the real topological string at all orders without new obstructions or anomalies.

What would settle it

A numerical mismatch between the Stokes constants extracted from the trans-series and the known integer disk invariants for the real topological string on local P2 would disprove the identification.

read the original abstract

We study the resurgent structure of Walcher's real topological string on general Calabi-Yau manifolds. We find trans-series solutions to the corresponding holomorphic anomaly equations, at all orders in the string coupling constant, by extending the operator formalism of the closed topological string, and we obtain explicit formulae for multi-instanton amplitudes. We find that the integer invariants counting disks appear as Stokes constants in the resurgent structure, and we provide experimental evidence for our results in the case of the real topological string on local $\mathbb{P}^2$.

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 / 1 minor

Summary. The manuscript studies the resurgent structure of Walcher's real topological string on general Calabi-Yau threefolds. It extends the operator formalism of closed topological strings to construct trans-series solutions to the associated holomorphic anomaly equations at all orders in the string coupling, derives explicit multi-instanton amplitudes, identifies the integer invariants counting disks as Stokes constants, and supplies experimental evidence for the local P² case.

Significance. If the extension of the closed-string operator formalism is valid without new anomalies or obstructions induced by the real involution, the identification of disk invariants with Stokes constants would furnish a direct non-perturbative link between open-string data and the Borel resummation of the real topological string partition function, strengthening the connection between resurgence techniques and topological string theory.

major comments (1)
  1. [Derivation of trans-series solutions and multi-instanton amplitudes] The central claim that the trans-series solutions reproduce the resurgent structure of the real topological string at all orders rests on the direct extension of the closed-string operator formalism without additional anomaly terms from the real involution. The manuscript supplies explicit multi-instanton formulae and experimental checks only for local P²; a general justification that no new obstructions appear at higher orders in the string coupling is required, as this assumption is load-bearing for the identification of disk invariants as Stokes constants on arbitrary Calabi-Yau threefolds.
minor comments (1)
  1. [Introduction and setup] Notation for the real involution and the associated holomorphic anomaly equations could be introduced more explicitly before the extension is applied, to aid readers unfamiliar with Walcher's construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its potential significance. We address the major comment below in a point-by-point manner.

read point-by-point responses

  1. Referee: The central claim that the trans-series solutions reproduce the resurgent structure of the real topological string at all orders rests on the direct extension of the closed-string operator formalism without additional anomaly terms from the real involution. The manuscript supplies explicit multi-instanton formulae and experimental checks only for local P²; a general justification that no new obstructions appear at higher orders in the string coupling is required, as this assumption is load-bearing for the identification of disk invariants as Stokes constants on arbitrary Calabi-Yau threefolds.

    Authors: We thank the referee for this observation. The extension proceeds by incorporating the real involution directly into the closed-string operator algebra and the associated recursive solution of the holomorphic anomaly equations. Because the real topological string is constructed to obey the identical anomaly equations (with the open sector entering solely through modified boundary conditions at the level of the disk invariants), no additional anomaly terms arise at any order in the string coupling. The multi-instanton amplitudes are obtained from the same recursive action of the operators used in the closed case, and the identification of the disk invariants with Stokes constants follows formally from the resulting trans-series structure. This derivation is independent of any specific Calabi-Yau and relies only on the general properties of the anomaly equations and the involution; the local P² checks serve as numerical verification rather than a limitation of the argument. We therefore maintain that the general justification is already contained in the formal construction presented in the manuscript. revision: no

Circularity Check

0 steps flagged

No significant circularity: derivation starts from holomorphic anomaly equations and extends prior operator formalism with independent experimental check

full rationale

The paper obtains trans-series solutions and multi-instanton formulae by extending the closed-string operator formalism to the real case and solving the holomorphic anomaly equations. Disk invariants are identified as Stokes constants via this explicit construction rather than by fitting or redefinition. No quoted step reduces a prediction to a fitted input by construction, nor does any load-bearing premise collapse to a self-citation chain that is itself unverified. The local P2 numerical evidence supplies an external benchmark. The extension assumption is stated openly but does not make the reported formulae tautological with the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on the holomorphic anomaly equations and the operator formalism of closed topological strings as background; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Holomorphic anomaly equations govern the dependence of topological string amplitudes on moduli.
    Invoked as the starting point for the trans-series solutions.
  • domain assumption The operator formalism developed for closed strings extends to the real case.
    Central technical step stated in the abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    Topological string partition function on CY threefolds factors into conifold terms powered by sheaf invariants, enabling non-perturbative Borel-resummed expression whose jumps are controlled by genus-zero GV invariant...

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