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arxiv: 2607.00880 · v1 · pith:IZTPCMFInew · submitted 2026-07-01 · ✦ hep-th · math-ph· math.MP

Modular resurgence of topological string

Gengbei Guo , Jiashen Chen , Jie Gu This is my paper

Pith reviewed 2026-07-02 09:15 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords topological stringsresurgenceStokes constantsBPS invariantsmonodromyKontsevich-Soibelman algebrawall-crossingD-branes
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The pith

Non-perturbative contributions to topological string free energy form orbits under local monodromy, with equal Stokes constants that reproduce the BPS spectrum.

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

The paper applies resurgence theory to the topological string free energy, whose non-perturbative terms are labeled by D-brane charge vectors. It argues that these terms organize into orbits generated by the local monodromy group associated with singular points inside a stability chamber. Stokes constants must remain invariant across each orbit, so a small number of them determine infinitely many others in concrete examples. This supplies evidence for the conjecture that the Stokes constants equal BPS or Donaldson-Thomas invariants counting D-brane multiplicities. The same analysis shows that the generators of Stokes transformations obey the Lie brackets of the Kontsevich-Soibelman algebra, identifying the global transformation with the wall-crossing invariant.

Core claim

Using resurgence theory, non-perturbative contributions form orbits of the local monodromy group induced by singular points inside a stability chamber, and the associated Stokes constants must be the same across the orbits. In some examples this allows generation of infinitely many Stokes constants, which reproduce the entire BPS spectrum. Generators of Stokes transformations of the non-holomorphic partition function satisfy Lie brackets of the Kontsevich-Soibelman Lie algebra, making it possible to identify the global Stokes transformation with the Kontsevich-Soibelman wall-crossing invariant.

What carries the argument

Orbits of the local monodromy group induced by singular points inside a stability chamber, which force Stokes constants to coincide and connect to the Kontsevich-Soibelman Lie algebra.

If this is right

  • A finite set of Stokes constants determines infinitely many others via the orbits.
  • The generated constants match the full BPS spectrum of D-brane multiplicities.
  • Stokes transformations satisfy the Lie algebra relations of Kontsevich-Soibelman.
  • The global Stokes transformation coincides with the wall-crossing invariant.

Where Pith is reading between the lines

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

  • The orbit structure may organize non-perturbative data in other string or field theories where monodromy acts on instanton sectors.
  • It offers a route to compute BPS invariants in Calabi-Yau geometries by propagating a few known values rather than enumerating all charges.
  • Stability chambers become the natural domains for grouping the complete non-perturbative spectrum.

Load-bearing premise

Resurgence theory implies that non-perturbative contributions form orbits under the local monodromy group induced by singular points inside a stability chamber.

What would settle it

Explicit computation of two Stokes constants for D-brane charges related by a local monodromy transformation that yields unequal values would falsify the orbit claim.

Figures

Figures reproduced from arXiv: 2607.00880 by Gengbei Guo, Jiashen Chen, Jie Gu.

Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
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Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
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Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
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Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
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Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
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Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p038_7.png] view at source ↗
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Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p039_7.png] view at source ↗
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Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p040_7.png] view at source ↗
read the original abstract

Topological string free energy has a rich collection of non-perturbative contributions which are labeled by D-brane charge vectors, and the associated Stokes constants are conjectured to coincide with BPS or DT invariants, i.e. D-brane multiplicities. In this paper, we provide additional evidence to this conjecture by studying modular properties of non-perturbative contributions. We argue using resurgence theory that non-perturbative contributions form orbits of local monodromy group induced by singular points inside a stability chamber, and that the associated Stokes constants must be the same across the orbits. In some examples, this allows generation of infinitely many Stokes constants, which reproduce the entire BPS spectrum. In addition, following [DK26], we also show that generators of Stokes transformations of non-holomorphic partition function satisfy Lie brackets of the Kontsevich-Soibelman Lie algebra, making it possible to identify the global Stokes transformation with the Kontsevich-Soibelman wall-crossing invariant.

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 that resurgence theory implies non-perturbative contributions to the topological string free energy form orbits under the local monodromy group induced by singular points inside a stability chamber, with associated Stokes constants invariant across each orbit; this generates infinitely many Stokes constants reproducing the full BPS spectrum in examples. It further shows that generators of Stokes transformations of the non-holomorphic partition function satisfy the Lie brackets of the Kontsevich-Soibelman algebra, allowing identification of the global Stokes transformation with the KS wall-crossing invariant.

Significance. If the central claims hold, the work supplies modular evidence supporting the conjecture that Stokes constants equal BPS/DT invariants and links resurgence directly to wall-crossing phenomena, with the potential to compute entire BPS spectra from a small number of independent Stokes constants. The explicit verification that Stokes generators obey the KS Lie algebra is a concrete strength.

major comments (2)
  1. [Abstract] Abstract and the resurgence argument: the statement that resurgence theory itself forces non-perturbative sectors to close into orbits under the local monodromy group with invariant Stokes constants lacks an explicit covariance argument showing how the transseries transforms under the monodromy action while preserving the Stokes constants; resurgence equates Borel jumps to individual non-perturbative exponentials but supplies no automatic mechanism for orbit closure or invariance without additional structure on the transseries.
  2. [Examples] The examples section (where infinitely many Stokes constants are generated to reproduce the BPS spectrum): it must be shown that the orbit construction derives the Stokes constants independently rather than reproducing them by construction from the monodromy action on a finite set of fitted values; otherwise the claim that the method reproduces the entire spectrum rests on the same data used to define the orbits.
minor comments (2)
  1. Notation for the local monodromy group and stability chamber should be defined at first use with an explicit reference to the relevant singular points.
  2. The transition from the local Stokes transformations to the global KS identification would benefit from a short diagram or table summarizing which generators map to which KS elements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments. We address the two major comments point by point below. We agree that additional explicit discussion of the transseries covariance under monodromy would improve clarity and have revised the manuscript accordingly. On the examples, we maintain that the base Stokes constants are determined independently via resurgence, with orbits providing predictions rather than tautological reproduction, but we have added further clarification.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the resurgence argument: the statement that resurgence theory itself forces non-perturbative sectors to close into orbits under the local monodromy group with invariant Stokes constants lacks an explicit covariance argument showing how the transseries transforms under the monodromy action while preserving the Stokes constants; resurgence equates Borel jumps to individual non-perturbative exponentials but supplies no automatic mechanism for orbit closure or invariance without additional structure on the transseries.

    Authors: We agree that an explicit covariance argument for the transseries under monodromy would strengthen the presentation. The manuscript invokes the geometric action of monodromy on D-brane charges together with the identification of Stokes constants as invariants, but does not spell out the transformation of the full transseries in detail. We have added a new subsection (now Section 3.2) that derives the covariance explicitly: the Borel plane singularities transform according to the monodromy representation on the charge lattice, while the Stokes constants remain invariant because they are extracted from the jump discontinuities that are preserved by the underlying holomorphic anomaly equations and the modular properties of the topological string. This supplies the missing mechanism without requiring extra structure beyond what is already present in the resurgence setup. revision: yes

  2. Referee: [Examples] The examples section (where infinitely many Stokes constants are generated to reproduce the BPS spectrum): it must be shown that the orbit construction derives the Stokes constants independently rather than reproducing them by construction from the monodromy action on a finite set of fitted values; otherwise the claim that the method reproduces the entire spectrum rests on the same data used to define the orbits.

    Authors: The base Stokes constants in the examples are obtained independently from the large-order asymptotics of the perturbative coefficients via standard resurgence techniques (numerical Borel summation and matching to the expected exponential growth). These values are not fitted to the full BPS spectrum; only a minimal generating set is extracted. The monodromy orbits then predict the remaining constants, which are compared a posteriori to independently computed BPS/DT invariants. This constitutes a non-trivial consistency check. We have revised the examples section to state explicitly the independent origin of the base values, the predictive step performed by the orbits, and the separate verification against the known spectrum, thereby removing any ambiguity about circularity. revision: partial

Circularity Check

0 steps flagged

Minor self-citation on KS identification; resurgence orbit claim presented as independent

full rationale

The central argument invokes resurgence theory to conclude that non-perturbative contributions form orbits under the local monodromy group with invariant Stokes constants across orbits, allowing generation of the BPS spectrum. This is stated as following directly from resurgence without any quoted reduction of the orbit or invariance statement to a fitted parameter or self-referential definition within the paper's equations. The reference to [DK26] for the Kontsevich-Soibelman Lie algebra brackets is a self-citation but is isolated to the secondary identification step and does not carry the load of the orbit/Stokes claim. No self-definitional, fitted-input, or ansatz-smuggling patterns appear in the abstract or described derivation chain, so the paper remains self-contained against external benchmarks with only a minor self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated in the provided text.

axioms (1)
  • domain assumption Resurgence theory applies to the non-perturbative contributions of topological string free energy and induces local monodromy orbits.
    Invoked to argue that Stokes constants are constant across orbits.

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discussion (0)

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