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Each conformal block appears in the Stokes phenomenon of the other, with exact formulas, and z-monodromy is fixed by alien calculus in the Borel plane.

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The two conformal blocks of the phi21 four-point function are each other's Stokes corrections in the large-C expansion, and their z-monodromy is governed by alien calculus in the Borel plane.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection A genuinely useful paper that makes the resurgent relation between the two phi21 conformal blocks explicit and checkable in Coulomb gas language; the central Stokes formulas hold up, but Lemma 3.1 is load-bearing and only sketched. the 4 major comments →

arxiv 2508.19059 v1 pith:C732EZWF submitted 2025-08-26 hep-th math-phmath.MP

Resurgence for large $c$ expansion in Coulomb gas formalism

Yong Li , Hongfei Shu This is my paper
classification hep-th math-phmath.MP MSC 81T4034M60
keywords resurgencealien calculusconformal blockslarge central charge expansionCoulomb gas formalismBorel-Laplace transformStokes phenomenonmonodromy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

At large central charge C, the genuinely non-perturbative information in a two-dimensional conformal field theory is not hidden inside one conformal block: it is stored in the other block. The paper proves this for the four-point function of degenerate operators φ2,1, whose two conformal blocks I1(C,z) and I2(C,z) are Coulomb-gas contour integrals. Using exact Borel–Laplace representations, it shows that the Stokes discontinuity of I2 as C crosses the imaginary axis is exactly I1 plus known self-corrections, and symmetrically that I1's Stokes jump is built from I2, in the explicit formulas (60) and (68). It then derives the monodromy of the pair around z=0 and z=1 from alien calculus in the Borel variable dual to C (Prop. 4.1) and exhibits a z→1−z duality of the Borel germs. If correct, the asymptotic expansion of one block can be resummed to discover the other internal operator, confirming a heuristic already proposed in the literature.

Core claim

The central claim is that I1(C,z)=∫_1^∞ Q^C dw and I2(C,z)=∫_0^z Q^C dw, with Q=w(w−1)(w−z), are not independent asymptotic objects. After writing each as a Laplace transform of a Borel germ—I2=Q_−^C L_0 Ψ(Q_−e^{−ζ}) and I1=Q_+^C (e^{−2πiC}/(1−e^{−2πiC})) L_{π/2+ε} Φ(Q_+e^{−ζ})—the paper computes all alien derivatives of the two germs and finds, in particular, Δ^+_{ζ^+_m} Ψ(Q_−e^{−ζ}) = Φ(Q_+e^{−ζ}). Consequently, crossing the Stokes ray with Im C<0, I2 is exactly Q_−^C (1+e^{−2πiC})/(1−e^{−2πiC}) L_{π/2+ε} Ψ(Q_−e^{−ζ}) + I1(C,z) (Theorem 3.5, eq. (60)), while I1's Stokes jump is expressed through I2 with the prefactors sin(2πC)/sin(3πC) and −sin(πC)/sin(3πC) (eq. (68)). The paper further es

What carries the argument

The machinery is a Borel–Laplace representation of the contour integrals I1 and I2 in the variable C. The three roots w0, w1, w2 of Q=w(w−1)(w−z) define two Borel germs: Ψ(Q)=Q·(2w1−w0−w2)/((w0−w1)(w0−w2)(w1−w2)) and Φ(Q)=Q·(w1+w2−2w0)/((w0−w1)(w0−w2)(w1−w2)). After substituting Q=Q_±e^{−ζ}, these become bψ(ζ) and bφ(ζ), whose only singularities are claimed to be integrable double branches on the two lattices ζ^−_m=2πim and ζ^+_m=−log(Q_+/Q_−)+2πim (with a mirrored lattice for bφ). The alien derivative Δ^+_ω measures the difference between two analytic continuations around a singularity, and the Stokes automorphism sums all lattice contributions; the paper's core computation is that the alie

Load-bearing premise

The load-bearing premise is Lemma 3.1: the Borel germs bψ and bφ have exactly the listed singular points—the two lattices {2πim} and {−log(Q_+/Q_−)+2πim}, plus the mirrored lattice for bφ—and each is an integrable double branch. All alien values, both Stokes formulas, and the monodromy matrices follow from this inventory; an extra singularity or a non-integrable branch would add terms everywhere.

What would settle it

Compute the Borel transform of the 1/C expansion of I2 at z=1/2 to high order and scan the Borel ζ-plane: Lemma 3.1 predicts singularities only at iπZ, all of square-root type. A singularity at any other point, or a logarithmic (non-integrable) branch at any lattice point, would invalidate formulas (60) and (68). Alternatively, numerically continue the hypergeometric expression for I2 around z=1 and compare with the monodromy matrix of Prop. 4.1; a mismatch in any matrix entry would localize the failure.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The large-C expansion of I2 is incomplete without I1: the full trans-series (61) contains three towers of terms, two of which are self-corrections of I2 and one of which is exactly the I1 tower.
  • Starting from the perturbative series of either block, Borel resummation along the imaginary direction reconstructs the other block's non-perturbative contribution.
  • The z-monodromy of (I1,I2) is determined by alien calculus in the Borel plane: Prop. 4.1 gives exact 2×2 monodromy matrices around z=0 and z=1, computable purely from how the Borel singular lattice moves.
  • The duality bψ(ζ,1−z)=−bφ(ζ,z) converts crossing symmetry into a sign flip of the Borel germ, so the full four-point function can be organized as a trans-series respecting z→1−z.
  • The same Coulomb-gas framework admits extensions to higher-point functions with more general degenerate operators, making Borel–Laplace analysis plus alien calculus a general resummation scheme for conformal-block data.

Where Pith is reading between the lines

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

  • An immediate corollary not spelled out in the paper is a computational bootstrap: if one conformal block is known to high order, the Stokes formulas give the other block's trans-series with the same z-dependent coefficients, without independent computation of the second block.
  • For polynomials Q(w) of degree five or higher, closed root formulas do not exist, so the explicit w_i construction stops; the paper's saddle-point/WKB remarks suggest that the alien lattice may be governed by the critical values Q_± rather than root formulas, so the same mutual-Stokes mechanism could survive in generalized integrals ∫_Γ Q^C dw.
  • One testable consequence of the singularity inventory is that in unitary or Liouville-type regimes, if additional Borel singular points appear, the exact relations (60) and (68) would acquire extra terms; the framework predicts which monodromy matrix entries would then be corrected.
  • The monodromy–alien calculus link, read as parametric resurgence, implies that the BPZ equation viewed as a Schrödinger equation with 1/C as the Planck constant should have WKB Stokes graphs whose crossing data match (60) and (68); verifying this would connect two independent resummation traditions.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper studies the large-C expansion of the two Coulomb-gas conformal blocks I1(C,z) and I2(C,z) for the four-point function of degenerate fields phi_{2,1}. It rewrites the cubic-root solutions w_i(Q,z), obtains exact Borel-Laplace representations of I1 and I2 (Propositions 2.1 and 2.2), identifies the Borel singularities, and computes alien derivatives of the Borel germs. The central results are the Stokes formulas (60) and (68), in which I1 appears as a Stokes correction to I2 and vice versa, a z -> 1-z duality of the Borel germs (Proposition 3.1), and a derivation of z=0,1 monodromy from motion of Borel singularities (Lemma 4.3, Proposition 4.1). Independent checks are provided via hypergeometric identities and numerical coefficient comparisons.

Significance. If the Borel-plane analysis is made fully rigorous, the paper provides a concrete, well-illustrated example of resurgence and parametric resurgence in 2D CFT, connecting the Coulomb gas integral representation to Stokes phenomena and z-monodromy. The explicit verification against hypergeometric identities, e.g. (63), (69), (70), and the numerical coefficient comparisons in Fig. 3 and Remark 3.6 are genuine strengths. The paper is a useful complement to Refs. [18,20] and a potential entry point for readers interested in co-equational resurgence. The conceptual claim that resurgence 'discovers' new conformal blocks is heuristic: both blocks are present from the beginning, and the main result is a structural relation between known objects rather than a prediction of new operator content.

major comments (4)
  1. [Section 3, Lemma 3.1 (Eqs. (43)-(44))] The exact and complete singularity inventory for bpsi and bphi is load-bearing for the alien calculus, but its proof is only sketched. Lemma 2.2 proves the integrable double-branch behavior for the specific germ H=(w1-w0)^{-1}(w1-w2)^{-1} at Q+; Lemma 3.1 extends this to the rational combinations Psi and Phi at both Q+ and Q- by saying 'mimicking the analysis'. Since the alien values in Lemma 3.2, equation (48), and hence equations (60), (68), and Lemma 4.3 depend on there being no additional singular points and on the leading (Q-Q_pm)^{1/2} behavior, the authors should provide computations for Psi at Q- and Phi at Q+ and Q- analogous to equations (27)-(30), including a demonstration that the relevant denominators do not vanish away from the critical values. An extra pole or a different leading power would change the factor 2 in (48) and the Stokes coefficients. This is the main gap in a
  2. [Section 3.1, Lemma 3.8 and Eq. (68)] The reverse alien derivatives in (65) are said to be obtained by inverting the Stokes automorphism Delta/+ via equation (56), but the inversion is not displayed. Because the final Stokes formula (68) for I1 depends on the m mod 3 structure and on the prefactors sin(2pi C)/sin(3pi C) and sin(pi C)/sin(3pi C), the derivation of the individual Delta^-_omega coefficients should be given, either by a direct computation or by an explicit inversion of the generating-function identity. The hypergeometric identity (69) is a useful external check, but it does not replace the missing derivation within the alien-calculus framework.
  3. [Sections 2.2 and 3.2, branch choices] The Borel-Laplace representations (35) and the Stokes formulas (60) and (68) involve phases such as e^{pm 2pi i C}, e^{pi i C}, and powers Q_+^C, Q_-^C, but the branch conventions are not fixed before they are used. The 'canonical choice' (77) appears only in Section 3.2, after the main Stokes formulas have been stated. Please state the branch choices for log Q_+ and log Q_- and for Q^C along the relevant contours from the outset, and verify that all prefactors are consistent with those choices and with the analytic continuation in z used in Section 4.
  4. [Abstract and Introduction] The paper repeatedly says that resurgence 'enables us to discover other internal operators (conformal blocks)'. In the body, both I1 and I2 are introduced as known integrals at the beginning, and the Stokes formula (60) is checked against the known hypergeometric identity (63). The result is a valuable structural connection, but the 'discovery' language overstates the novelty and should be qualified, e.g. by saying that resurgence reproduces and relates known conformal blocks in this explicit example.
minor comments (6)
  1. [Section 3 heading] Typo: 'Alien culculus' should be 'Alien calculus'.
  2. [Figure 4 caption] Typo: 'locaeed' should be 'located'.
  3. [Lemma 3.8, Eq. (65)] The indexing of the last case, 'else (m >= 0)', should be checked; in particular, the m=0 entry appears to be included in the 'else' branch but this is not explicitly stated.
  4. [Remark 3.9] The phrase 'formula (3.9)' likely refers to equation (68) of this version; please correct the cross-reference.
  5. [Appendix A, Lemma A.1 proof] Minor typo: 'The Let y = arccos(x)' should read 'Let y = arccos(x)'.
  6. [Eq. (15)] The word 'possibly' is appropriate, but it should be reconciled with the definitive singularity inventory claimed in Lemma 3.1; the reader should know which singularities are excluded and why.

Circularity Check

0 steps flagged

No significant circularity: the Stokes and monodromy results are derived from Borel-plane monodromy of the explicit roots w_i, with hypergeometric identities used only as external cross-checks.

full rationale

The paper's central derivation chain is not circular. Propositions 2.1 and 2.2 obtain the Borel-Laplace representations (20) and (35) by direct contour rewriting of the defining integrals, using the explicit Cardano roots w_i (12) and the analytic continuation of arccos (Lemma A.2). The load-bearing alien values in Lemma 3.2, such as Δ_{ζ+_m} bψ = bφ and Δ_{ζ−_m} bψ = 2 bψ, are computed from the branch monodromy of w_i (equations (16)-(18) and (49)-(50)), not assumed from the Stokes relations being proved. Theorem 3.5 (60) and eq. (68) then follow by applying the symbolic Stokes automorphism (51)-(56); the appearance of I1 in the Stokes discontinuity of I2 is an algebraic consequence of the computed alien derivative plus the Laplace representation of I1, not a restatement of the definition of I1. The hypergeometric identities (63) and (69) are used only as external cross-checks in Remarks 3.7 and 3.9, not as inputs to the derivation. No fitted parameter is renamed as a prediction. The only load-bearing point with terse proof is Lemma 3.1's extension of the complete integrable singularity lattice to Φ and to Ψ at Q_- via 'mimicking the analysis' of Lemma 2.2; this is a rigor/completeness concern that could affect correctness if the singularity inventory were incomplete, but it is an unproven assertion about the same germs, not a circular reduction to the target result. The one self-citation, [30], supports a technical homotopy/vector-field step in Lemma 4.3 and is not load-bearing for the main claim. Hence no circularity is present.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The construction introduces no fitted parameters and no invented physical entities. The load-bearing inputs are: (i) the Dotsenko-Fateev dictionary identifying the contour integrals with conformal blocks, (ii) standard resurgence machinery, (iii) the exact singularity inventory of the Borel germs (Lemma 3.1), which is partly sketched, and (iv) branch conventions for analytic continuation in z (eq (77)). These carry the central claim; the w_i parametrization (12) is standard math.

axioms (5)
  • domain assumption Coulomb gas integrals I1(C,z) = integral_1^infty Q^C dw and I2(C,z) = integral_0^z Q^C dw are the conformal blocks of the phi21 four-point function in the channels of internal operators phi11 and phi31
    Imported from Dotsenko-Fateev [21,22]; the physics identification (operator content, charge neutrality, contour prescriptions) is prior literature, not derived here (Section 1, eqs (1)-(4)).
  • standard math Borel-Laplace and resurgence machinery: singular points of the Borel germs are integrable double-branch, alien derivatives are computed as analytic-continuation differences, and Stokes automorphisms satisfy L_{d-epsilon} = L_{d+epsilon} composed with Delta-plus_d
    Standard resurgence theory [4,27], cited and used throughout Sections 2-3.
  • ad hoc to paper The singular set of the Borel germs bpsi and bphi is exactly {zeta_m^- = 2pi i m} union {zeta_m^+ = -log(Q+/Q-) + 2pi i m} (resp. shifted set), all integrable double-branch
    Lemma 3.1: explicit for Psi via Lemma 2.2, but asserted by 'mimicking' for Phi and for integrability of all lattice points; the alien values in Lemma 3.2 depend entirely on this.
  • ad hoc to paper Branch choices Q_+^C(1-z) = e^{pi i C} Q_-^C(z) and Q_-^C(1-z) = e^{-pi i C} Q_+^C(z) (eq (77))
    Called the 'canonical choice' in the paper; signs in Proposition 3.2 depend on it. Stated without a full derivation from the z-branch structure.
  • domain assumption Analytic continuation in z beyond a neighborhood of 1/2 preserves the stated singularity structure and the Stokes formulas
    Section 2 opening says 'Other regions of z can be obtained via the analytical continuation'; used for Props 3.2 and 4.1 without delimiting the domain.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Resurgence for large $c$ expansion in Coulomb gas formalism." pith.science (2026). https://pith.science/paper/C732EZWF

@misc{pith2026250819059,
  author       = {Pith},
  title        = {Pith review of: Resurgence for large $c$ expansion in Coulomb gas formalism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C732EZWF}},
  note         = {Machine review of arXiv:2508.19059}
}
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abstract

We develop a resurgence analysis for large central charge (large $C$) expansions in two-dimensional CFTs using the Coulomb gas formalism. Through the exact Borel-Laplace representations of the conformal blocks $I_1(C,z)$ and $I_2(C,z)$ associated with the four-point correlation function $ \langle \phi_{2,1}(0)\phi_{2,1}(z,\bar{z})\phi_{2,1}(1)\phi_{2,1}(\infty)\rangle$, we demonstrate that $I_1(C,z)$ participates in the Stokes phenomenon of $I_2(C,z)$ (and vice versa), and establish that monodromy in $z$ arises from alien calculus in the Borel plane variable $\zeta$ (Borel dual to $C$). From a given conformal block, resurgence theory thus enables us to discover other internal operators (conformal blocks). This approach establishes a non-perturbative connection between conformal blocks, shedding light on the resurgence phenomena in more general quantum field theories.

Figures

Figures reproduced from arXiv: 2508.19059 by Hongfei Shu, Yong Li.

Figure 1
Figure 1. Figure 1: When z lies in a small neighborhood of 1 2 , the cubic projection Q(w) = w(w−1)(w− z) induces a local triple cover. The fibers are parameterized by three maps wi (i = 0, 1, 2). Proposition 2.1 expresses the integral I2(C, z) as an integral over the Q-plane along the blue line segments, which are preimages of w0 and w2. Similarly, Proposition 2.2 represents I1(C, z) via integration over the Q-plane along th… view at source ↗
Figure 2
Figure 2. Figure 2: Writing the integral along the real axis as an integral on the sum of Hankel contours [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: We compare sn and s ′ n at z = 1/2 for pure negative imaginary C, where iC is pure real positive. 3 Alien culculus on Ψ(Q−e −ζ ) and Φ(Q+e −ζ ) What we have shown are the Borel-Laplace expressions of I1 and I2 for z near 1 2 . In this sec￾tion, we discuss the alien calculus of the Borel germs used in Proposition 2.1 and Proposition 2.2. For abbreviation, we use ψb(ζ) := Ψ(Q−e −ζ ) and φb(ζ) := Φ(Q+e −ζ ). … view at source ↗
Figure 4
Figure 4. Figure 4: For a point ω locaeed in direction d, the path used in cont+ is the black curve from a point near 0 to a point near ω. + means going from right. Thus ∆+ is an operator that maps a germ at εeid (for ε ≪ 1) to a germ holomorphic at the same base point. To conclude this brief introduction to the operator ∆+ ω , we emphasize— though we will not use this property in the present paper—that it satisfies a general… view at source ↗
Figure 5
Figure 5. Figure 5: When z lies in a neighborhood of 1 2 , the singular set of ψb is depicted in the left panel, while that of φb appears in the right panel. They are both lattice πiZ as z = 1 2 . Lemma 3.3. Recall ψb in equation (42). As z near 1 2 , we have ∆/ + π 2 ψb = ψb + 2 X m≥1 e −2πimCψb +  Q+ Q− C X m≥1 e −2πimCφ. b (57) Proof. This is a result of the definition of ∆/ + π 2 in equation (54) and alien calculus in L… view at source ↗
Figure 6
Figure 6. Figure 6: When z is near 1 2 , as in Proposition 2.1, ψb as singular points shown in the left picture. The middle picture is the integral curve that we use when we perform analytic continuation in the variable C from ReC > 0 to ImC < 0. These integrals (except for the green one) provide the Stokes terms along the (almost) π 2 direction. Since every singular point is integrable, these integrals on the Hankel contours… view at source ↗
Figure 7
Figure 7. Figure 7: In the above left scope, z starts at 1 2 , goes along the real axis to a point near 0, circles around 0 negatively and goes back to 1 2 . The corresponding log( Q+ Q− ) runs along the blue line in the above right scope. As z goes around 1 in the below left scope, the corresponding trace of log( Q+ Q− ) is shown in the below right scope. Let ∆z=pf(z) := f(z) − contz=pf(z) (83) be the difference of two sheet… view at source ↗
Figure 8
Figure 8. Figure 8: In the first row, as z goes around 0 negatively in top left picture, the singular points of ψb move along the blue curves from ζ + m to ζ + m−2 in the top middle picture. Correspondingly, the singular points of φb move along the blue curves from −ζ + −m to −ζ + −(m+2), as shown in the top right picture. In the second row, as z goes around 1 negatively in the bottom left picture, the singular points of ψb m… view at source ↗
Figure 9
Figure 9. Figure 9: Computation of ∆z=1L π 2 +εφb We then compute the difference of two Laplace transforms as shown in the middle picture. The result is ∆z=1L π 2 +εφb = L π 2 +εφb − contz=1L π 2 +εφb =  − Z γ1 + Z γ2  e −Cζφdζ, b (88) where the paths γ1 and γ2 are shown in the right picture. The symbols we used in the last term of above formula are by the analytic continuation of germ φb at ζ − m for all m ∈ Z: contζ −m φb… view at source ↗

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Works this paper leans on

31 extracted references · 18 canonical work pages · 3 internal anchors

  1. [1]

    Beisert et al., Review of AdS/CFT Integrability: An Overview , Lett

    N. Beisert et al., Review of AdS/CFT Integrability: An Overview , Lett. Math. Phys. 99 (2012) 3–32, [ 1012.3982]

    Pith/arXiv arXiv 2012

  2. [2]

    Pestun et al., Localization techniques in quantum field theories , J

    V. Pestun et al., Localization techniques in quantum field theories , J. Phys. A 50 (2017) 440301, [1608.02952]

    Pith/arXiv arXiv 2017

  3. [3]

    J. M. Maldacena, The Large N limit of superconformal field theories and supergravity , Adv. Theor. Math. Phys. 2 (1998) 231–252, [ hep-th/9711200]

    Pith/arXiv arXiv 1998

  4. [4]

    Ecalle, Les fonctions r´ esurgentes

    J. Ecalle, Les fonctions r´ esurgentes. Tome I, vol. 5 of Publications Math´ ematiques d’Orsay 81 [Mathematical Publications of Orsay 81] . Universit´ e de Paris-Sud, D´ epartement de Math´ ematiques, Orsay, 1981

    work page 1981

  5. [5]

    Ecalle, Les fonctions r´ esurgentes

    J. Ecalle, Les fonctions r´ esurgentes. Tome II, vol. 6 of Publications Math´ ematiques d’Orsay 81 [Mathematical Publications of Orsay 81] . Universit´ e de Paris-Sud, D´ epartement de Math´ ematiques, Orsay, 1981

    work page 1981

  6. [6]

    Ecalle, Les fonctions r´ esurgentes

    J. Ecalle, Les fonctions r´ esurgentes. Tome III, vol. 5 of Publications Math´ ematiques d’Orsay 85 [Mathematical Publications of Orsay 85] . Universit´ e de Paris-Sud, D´ epartement de Math´ ematiques, Orsay, 1985

    work page 1985

  7. [7]

    F. J. Dyson, Divergence of perturbation theory in quantum electrodynamics , Phys. Rev. 85 (1952) 631–632. 24

    work page 1952

  8. [8]

    J.-C. L. Guillou and J. Zinn-Justin, Large-order behaviour of perturbation theory. North-Holland : Sole distributors in USA and Canada Elsevier Science Pub. Co., 1990

    work page 1990

  9. [9]

    Mari˜ no,Lectures on non-perturbative effects in large N gauge theories, matrix models and strings , Fortsch

    M. Mari˜ no,Lectures on non-perturbative effects in large N gauge theories, matrix models and strings , Fortsch. Phys. 62 (2014) 455–540, [ 1206.6272]

    Pith/arXiv arXiv 2014

  10. [10]

    Dorigoni, An Introduction to Resurgence, Trans-Series and Alien Calculus , Annals Phys

    D. Dorigoni, An Introduction to Resurgence, Trans-Series and Alien Calculus , Annals Phys. 409 (2019) 167914, [ 1411.3585]

    Pith/arXiv arXiv 2019

  11. [11]

    Aniceto, G

    I. Aniceto, G. Basar and R. Schiappa, A Primer on Resurgent Transseries and Their Asymptotics, Phys. Rept. 809 (2019) 1–135, [ 1802.10441]

    Pith/arXiv arXiv 2019

  12. [12]

    Di Francesco, P

    P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory . Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997, 10.1007/978-1-4612-2256-9

    doi:10.1007/978-1-4612-2256-9 1997

  13. [13]

    A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory , Nucl. Phys. B 241 (1984) 333–380

    work page 1984

  14. [14]

    Harlow, J

    D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory , JHEP 12 (2011) 071, [ 1108.4417]

    Pith/arXiv arXiv 2011

  15. [15]

    Dorn and H

    H. Dorn and H. J. Otto, Two and three point functions in Liouville theory , Nucl. Phys. B 429 (1994) 375–388, [ hep-th/9403141]

    Pith/arXiv arXiv 1994

  16. [16]

    A. B. Zamolodchikov and A. B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory , Nucl. Phys. B 477 (1996) 577–605, [hep-th/9506136]

    Pith/arXiv arXiv 1996

  17. [17]

    Resurgence in Liouville Theory

    N. Benjamin, S. Collier, A. Maloney and V. Meruliya, Resurgence in Liouville theory , JHEP 01 (2025) 038, [ 2408.14574]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  18. [18]

    Resurgence, Conformal Blocks, and the Sum over Geometries in Quantum Gravity

    N. Benjamin, S. Collier, A. Maloney and V. Meruliya, Resurgence, conformal blocks, and the sum over geometries in quantum gravity , JHEP 05 (2023) 166, [ 2302.12851]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  19. [19]

    M. V. Berry and C. J. Howls, Hyperasymptotics for integrals with saddles — Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, Proc. Roy. Soc. Lond. A V olume 434(1991) 657–675

    work page 1991

  20. [20]

    On the $1/c$ expansion in $2d$ CFTs with degenerate operators

    A. Bissi, N. Dondi, A. Piazza, T. Reis and M. Serone, On the 1/c expansion in 2d CFTs with degenerate operators , JHEP 03 (2025) 092, [ 2412.04387]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  21. [21]

    V. S. Dotsenko and V. A. Fateev, Conformal Algebra and Multipoint Correlation Functions in Two-Dimensional Statistical Models , Nucl. Phys. B 240 (1984) 312

    work page 1984

  22. [22]

    V. S. Dotsenko and V. A. Fateev, Four Point Correlation Functions and the Operator Algebra in the Two-Dimensional Conformal Invariant Theories with the Central Charge c < 1, Nucl. Phys. B 251 (1985) 691–734

    1985

  23. [23]

    Gu and B

    X. Gu and B. Haghighat, Liouville conformal blocks and stokes phenomena , Beijing J. of pure and appl. math 1 (2024) . 25

    work page 2024

  24. [24]

    Ecalle, Weighted products and parametric resurgence, Analyse algebrique des perturbations singulieres

    J. Ecalle, Weighted products and parametric resurgence, Analyse algebrique des perturbations singulieres. I (1994)

    work page 1994

  25. [25]

    T. Aoki, T. Takahashi and M. Tanda, The hypergeometric function, the confluent hypergeometric function and WKB solutions , Journal of the Mathematical Society of Japan 73 (2021) 1019 – 1062

    work page 2021

  26. [26]

    R. W. D. Nickalls, A new approach to solving the cubic: Cardan ’s solution revealed , The Mathematical Gazette 77 (1993) 354 – 359

    work page 1993

  27. [27]

    Mitschi and D

    C. Mitschi and D. Sauzin, Divergent series, summability and resurgence. I , vol. 2153 of Lecture Notes in Mathematics . Springer, [Cham], 2016, 10.1007/978-3-319-28736-2

    doi:10.1007/978-3-319-28736-2 2016

  28. [28]

    Sauzin, Resurgent functions and splitting problems , RIMS Kokyuroku 1493 (2005)

    D. Sauzin, Resurgent functions and splitting problems , RIMS Kokyuroku 1493 (2005)

    work page 2005

  29. [29]

    N. M. Temme, Asymptotic Methods for Integrals . WORLD SCIENTIFIC, 2014, 10.1142/9195

    doi:10.1142/9195 2014

  30. [30]

    Y. Li, D. Sauzin and S. Sun, On the Moyal Star Product of Resurgent Series , Annales de l’Institut Fourier 73 (2023) 1987–2027

    work page 2023

  31. [31]

    ’t Hooft, Can We Make Sense Out of Quantum Chromodynamics? , Subnucl

    G. ’t Hooft, Can We Make Sense Out of Quantum Chromodynamics? , Subnucl. Ser. 15 (1979) 943. 26

    work page 1979

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.