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arxiv: 2508.10112 · v3 · submitted 2025-08-13 · ✦ hep-th · math-ph· math.GT· math.MP· math.NT

c_(rm eff) from Resurgence at the Stokes Line

Griffen Adams , Ovidiu Costin , Gerald V. Dunne , Sergei Gukov , O\u{g}uz \"Oner This is my paper

Pith reviewed 2026-05-18 22:28 UTC · model grok-4.3

classification ✦ hep-th math-phmath.GTmath.MPmath.NT
keywords resurgenceChern-Simons theoryq-serieseffective central charge3d-3d correspondenceBPS statestransseriesfalse theta functions
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The pith

Resurgent cyclic orbits and one leading q-series term fix the large-order growth rate of dual coefficients and yield an effective central charge in related 3d N=2 theories.

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

The paper demonstrates that the algebraic structure of resurgent cyclic orbits, when paired with only the leading term of a q-series, is sufficient to determine the asymptotic growth rate of coefficients in the corresponding dual q-series. This growth exponent receives a Cardy-like interpretation as an effective central charge for a three-dimensional N=2 supersymmetric theory connected to Chern-Simons theory by the 3d-3d correspondence. A sympathetic reader would care because the result shows how resurgence can extract this physical quantity directly from path-integral data without requiring the full series or extra input, and it supplies a concrete way to continue q-series across their natural boundaries via decompositions into unary false theta functions.

Core claim

The algebraic structure of the resurgent cyclic orbits, combined with just the leading term of the q-series, completely determines the large order rate of growth of the dual q-series coefficients. The essential exponent of this asymptotic growth has a Cardy-like interpretation of an effective central charge in a 3 dimensional quantum field theory with N=2 supersymmetry related to the Chern-Simons theory through the 3d-3d correspondence.

What carries the argument

Resurgent cyclic orbits, whose algebraic structure together with the leading q-series term determines the precise asymptotic growth rate of the dual q-series coefficients.

If this is right

  • The large-order growth rate of dual q-series coefficients is fixed exactly by the resurgent orbit structure and one leading term.
  • The resulting exponent supplies the effective central charge c_eff for the associated 3d N=2 supersymmetric theory.
  • Transseries decompositions into unary false theta functions permit analytic continuation of the series across the natural boundary.
  • Resurgence supplies a direct link from Chern-Simons path integrals to BPS state counting in the dual 3d theory.

Where Pith is reading between the lines

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

  • The same orbit algebra might be applied to other families of integrals in Chern-Simons theory to predict growth rates without recomputing full series.
  • The method could be tested on explicit low-order terms of known dual q-series to verify the extracted central charge against independent 3d-3d calculations.
  • If the orbit structure is universal across related theories, the approach may give a uniform way to read off effective central charges for broader classes of N=2 theories.

Load-bearing premise

The algebraic structure of the resurgent cyclic orbits together with only the leading q-series term is sufficient to fix the precise large-order growth rate of the dual coefficients without additional terms or external data.

What would settle it

For a concrete Mordell-Borel integral arising in Chern-Simons theory, extract the predicted large-order growth exponent from the resurgent cyclic orbit algebra and the single leading term, then compare it directly to the numerically computed asymptotic growth of the coefficients in the dual q-series.

Figures

Figures reproduced from arXiv: 2508.10112 by Gerald V. Dunne, Griffen Adams, O\u{g}uz \"Oner, Ovidiu Costin, Sergei Gukov.

Figure 1
Figure 1. Figure 1: This log plot compares the predicted growth (4.41) and (4.45), respectively, of the coefficients of Φ(1) 7 (i √q) ∨ (solid red line) and Φ(2) 7 (−q) ∨ (solid green line), as in (4.7)-(4.8), to the coefficients determined by the numerical algorithm in [2], which are plotted as black points for both q-series. So in the limit t → 0 + limit, the finiteness of the integrals on the LHS of (4.35) implies that the… view at source ↗
Figure 2
Figure 2. Figure 2: This log plot compares the predicted growth of the coefficients of Φ(1) 9 (i √q) (solid red line) and Φ(2) 9 (−q) (solid green line), as in (4.7)-(4.8), to the coefficients determined by the numerical algorithm in [2], which are plotted as black points for Φ(1) 9 (i √q) and blue points for Φ(2) 9 (−q). Note the slow growth of the coefficients, compared to [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Values of ∆ep, the key parameter defined in (4.3) for the duals of false theta functions, for odd p values. This determines ceff as in (4.5). Note the wide variation of values. Smaller ∆ep corresponds to slower growth of the q-series coefficients. 4.6. Comparisons with other results for Duals of False Theta Functions. In this section we compare our new results for duals of false theta functions with other … view at source ↗
Figure 4
Figure 4. Figure 4: This log plot compares the predicted growth [solid red line] for the q-series coefficients of X (1) (2,3,5)(q) ∨ , as in (5.1)-(5.8), with the coefficients determined by the numerical algorithm in [2], which are plotted as black points. Comparing with the decomposition identities in (3.19) leads to the identifications X (1) (2,3,5)(q) ∨ = − 2 3 (χ0(q) − 2) = − 2 3 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: This log plot compares the predicted growth [solid red line] for the q-series coefficients of X (1) (2,3,7)(q) ∨ , as in (5.1)-(5.14), with the coefficients determined by the numerical algorithm in [2], which are plotted as black points. j Non-unary dual q-series X (j) (2,3,11)(q) ∨ 1 1 6 q −25/264 −1 + 15q + 65q 2 + 175q 3 + . . .  2 1 6 q −1/264 5 + 5q + 15q 2 + 60q 3 + 125q 4 + . . .  3 1 6 q −49/264 … view at source ↗
Figure 6
Figure 6. Figure 6: This log plot compares the predicted growth [solid red line] for the q-series coefficients of X (1) (2,3,11)(q) ∨ , as in (5.1), with the coefficients determined by the numerical algorithm in [2], which are plotted as black points. 5.4. New Results: Growth Rates for Σ(2, 3, 13). For p = 13, the integral vectors defined in (3.18) have (p−1) 2 = 6 components. Once again, unlike the p = 5 and p = 7 cases, for… view at source ↗
Figure 7
Figure 7. Figure 7: This log plot compares the predicted growth for the q-series coeffi￾cients of X (1) (2,3,13)(q) ∨ given in (5.1) for p = 13 and j = 1 (red solid line) to the numerically determined coefficients of X (1) (2,3,13)(q) ∨ (black points). 5.5. Comparisons with other results for Duals for Brieskorn Spheres. In this section, we compare our results for the effective central charges of the half indices c T[M3] eff, … view at source ↗
Figure 8
Figure 8. Figure 8: Values of ∆e(2,3,p) for the q-series associated to orientation-reversed Brieskorn spheres with p = 6k ± 1, with k ∈ Z +. Note the wide variation of values. Smaller ∆e(2,3,p) corresponds to slower growth of the q-series coefficients. arguments, and we comment on the comparison of our results. In addition, we use the methods outlined in Appendix B of [2] to expand our analysis to any orientation-reversed Bri… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison plots including subleading growth of Ψ(a) 7 (−q) ∨ for a = 2 (left) and a = 6 (right). In both figures, we plot the estimated growth without subleading corrections as a solid red line, and the estimated growth with subleading corrections as a solid blue line. Then, we plot every other coefficient of Ψ(a) 7 (−q) ∨ as black points. We see that subleading corrections are more pronounced for larger … view at source ↗
Figure 10
Figure 10. Figure 10: Comparison plots including subleading growth of X (j) (2,3,11)(q) ∨ for j = 1 (left) and j = 5 (right). In both figures, we plot the estimated growth without subleading corrections as a solid red line, and the estimated growth with subleading corrections as a solid blue line. Then, we plot every other coefficient (to avoid clutter) of X (j) (2,3,11)(q) ∨ as black points. We see that subleading corrections… view at source ↗
Figure 11
Figure 11. Figure 11: The growth is very slow for X (6) (2,3,13)(q) ∨ , since the ∆e(2,3,13) param￾eter is small: see [PITH_FULL_IMAGE:figures/full_fig_p041_11.png] view at source ↗
read the original abstract

In recent papers [1,2], a new method to cross the natural boundary has been proposed, and applied to Mordell-Borel integrals arising in the study of Chern-Simons theory, based on decompositions into {\it resurgent cyclic orbits}. Resurgent analysis on the Stokes line leads to a unique transseries decomposition in terms of unary false theta functions, which can be continued across the natural boundary to produce dual $q$-series whose integer-valued coefficients enumerate BPS states. This constitutes a deeper new manifestation of resurgence in quantum field theoretic path integrals. In this paper we show that the algebraic structure of the {\it resurgent cyclic orbits}, combined with just the leading term of the $q$-series, completely determines the large order rate of growth of the dual $q$-series coefficients. The essential exponent of this asymptotic growth has a Cardy-like interpretation [10] of an effective central charge in a 3 dimensional quantum field theory with $\mathcal{N}=2$ supersymmetry related to the Chern-Simons theory through the $3d$-$3d$ correspondence.

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 / 2 minor

Summary. The manuscript develops a resurgence approach to crossing natural boundaries in q-series arising from Mordell-Borel integrals in Chern-Simons theory. It constructs a unique transseries decomposition into unary false theta functions on the Stokes line from the algebraic structure of resurgent cyclic orbits; this decomposition, together with only the leading term of the original q-series, is shown to fix the large-order asymptotic growth rate of the coefficients in the dual q-series. The essential exponent in this growth admits a Cardy-like interpretation as an effective central charge c_eff in an N=2 supersymmetric 3d QFT related to the Chern-Simons theory by the 3d-3d correspondence.

Significance. If the central claim holds, the work supplies a concrete algebraic mechanism by which resurgence encodes physical data (the growth exponent) with minimal external input, furnishing a new link between resurgent transseries and BPS-state counting in supersymmetric theories. This strengthens the case for resurgence as a tool that extracts effective central charges directly from path-integral data.

major comments (1)
  1. Abstract and the paragraph beginning 'In this paper we show': the assertion that the algebraic structure of the resurgent cyclic orbits 'completely determines' the large-order growth rate is load-bearing for the main result. The manuscript must exhibit an explicit, self-contained derivation (or at least one fully worked non-trivial example) showing that no further coefficients or external data beyond the leading term are required; otherwise the claim reduces to a normalization statement rather than a determination of the exponent.
minor comments (2)
  1. The relation between the unary false theta functions appearing in the transseries and the integer-valued coefficients of the dual q-series should be stated with an explicit formula or generating-function identity, preferably in a dedicated subsection.
  2. Clarify whether the Cardy-like identification of the growth exponent with c_eff follows directly from the 3d-3d correspondence or requires additional input from reference [10]; a short paragraph contrasting the present derivation with the standard Cardy formula would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the work, and recommendation for minor revision. We address the single major comment below and have revised the manuscript accordingly to strengthen the presentation of the central claim.

read point-by-point responses

  1. Referee: Abstract and the paragraph beginning 'In this paper we show': the assertion that the algebraic structure of the resurgent cyclic orbits 'completely determines' the large-order growth rate is load-bearing for the main result. The manuscript must exhibit an explicit, self-contained derivation (or at least one fully worked non-trivial example) showing that no further coefficients or external data beyond the leading term are required; otherwise the claim reduces to a normalization statement rather than a determination of the exponent.

    Authors: We agree that an explicit illustration is valuable for making the claim fully transparent. The algebraic structure of the resurgent cyclic orbits determines the transseries parameters on the Stokes line, including the specific unary false theta functions and their associated singularity data. These parameters directly fix the exponential growth rate of the dual q-series coefficients through the resurgence analysis. The leading term of the original q-series enters only as an overall normalization factor for the asymptotic prefactor; the exponent itself is an invariant of the orbit algebra and does not depend on higher coefficients. In the revised manuscript we have added a self-contained derivation of this separation (new subsection in Section 4) together with a fully worked non-trivial example for a specific Mordell-Borel integral. In that example the growth exponent is computed using solely the orbit structure and the leading coefficient, reproducing the expected c_eff without any additional input, thereby confirming that the determination is not merely a normalization statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the large-order growth rate of dual q-series coefficients from the algebraic structure of resurgent cyclic orbits on the Stokes line together with the leading q-series term. This follows directly from the unique transseries decomposition into unary false theta functions, which supplies the required algebraic relations without reducing to a fitted input or self-citation chain. The leading term normalizes scale only, and the Cardy-like reading of the resulting exponent via the 3d-3d correspondence is an interpretive remark citing external work rather than a load-bearing step in the determination itself. The construction is self-contained against the stated inputs and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Assessment based solely on abstract; no explicit free parameters are stated. Implicit assumptions concern the applicability of resurgence to the relevant path integrals and the validity of the 3d-3d correspondence for the central-charge interpretation.

axioms (2)
  • domain assumption Resurgent analysis on the Stokes line yields a unique transseries decomposition into unary false theta functions that can be continued across the natural boundary
    Stated in the abstract as the foundation for producing the dual q-series.
  • domain assumption The 3d-3d correspondence relates the Chern-Simons theory to a 3d N=2 supersymmetric QFT in which the growth exponent has a Cardy-like meaning as effective central charge
    Invoked for the physical interpretation of the asymptotic exponent.

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Reference graph

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