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REVIEW 4 major objections 4 minor 36 references

Modular resurgent structures for vectors

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Vector-valued resurgence fully captures false theta functions: Stokes constants are rotated by the modular S-matrix, and median resummation reconstructs the Eichler integrals.

desk verdict The unary-theta half of this paper is a solid, fully worked advance; the advertised general weight-1/2 and 3/2 extension is overstated and does not meet the paper's own Definition 2.5. read the letter →

arxiv 2608.08902 v1 pith:XEOUDEQK submitted 2026-08-09 math.NT hep-thmath-phmath.CVmath.MP

Miranda C. N. Cheng , Ioana Coman , Veronica Fantini , Claudia Rella This is my paper · ORCID
classification math.NThep-thmath-phmath.CVmath.MP MSC 11F3711F27
keywords vector-valuedmodularresurgenceEichlerintegralsfalsethetafunctionsquantumformsmedianresummationStokesconstantsDirichletL-functions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Building on the scalar framework of modular resurgence, this paper extends it to vectors of asymptotic series, introducing vector-valued modular resurgent series (Definition 2.5). The paper proves that two natural families are vector-valued MRSs: vectors of q-Pochhammer symbols, which are a repackaging of scalar results, and vectors of Eichler integrals of unary theta series, which require the vector-valued framework because their Stokes constants are rotated by the modular S-matrix. For the latter, the paper establishes the full paradigm: the Borel transform has a single tower of simple poles, the Stokes constants form a Dirichlet vector that decomposes into L-functions, the vector is a quantum modular form for SL2(Z), and median resummation reconstructs the Eichler integrals from their divergent expansions. The unary theta case is the fully worked example of a general result outlined for vector-valued cusp forms of weight 1/2 and 3/2.

What carries the argument

Definition 2.5: a vector of Gevrey-1 series is a vector-valued MRS if its components share one tower of Borel singularities at $A m$, have trivial secondary resurgent series, and their Stokes constants form Dirichlet vectors that are linear combinations of a common vector of $L$-functions via matrices $M_\pm$. For the Eichler example, the key identity is the Borel transform formula $B[\tau^{3/2-\nu}\widetilde{\mathrm{EI}}_N^{(\nu)}](\xi) = -\frac{1}{2\pi i}(\frac{2N\xi}{\pi i})^{1/2-\nu} \sum_{m=1}^\infty \frac{m^{2\nu-1}\mathrm{St}_m^{(\nu)}}{\xi - \pi i m^2/(2N)}$, which manifestly exhibits the simple-pole tower and the Stokes vectors. In the $\nu=1$ case the $\sinh$-ratio kernel $K_{N,k}(x)=\sinh((N-k)x)/\sinh(Nx)$ gives a closed form for the Borel transform and shows geometrically how the poles of the kernel produce the Stokes constants.

What would settle it

Take a vector-valued cusp form $g$ of weight $3/2$ that is not a linear combination of unary theta series, compute its Stokes–Dirichlet vector from Eq. (228), and check whether its coefficients admit an Euler product; if they do not, part 3 of Definition 2.5 fails. A more direct test is to verify numerically whether the median resummation identity (235) holds for such $g$.

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Extended reading notes

Core claim

The central claim is that the vector $\mathrm{EI}_N^{(\nu)}$ of Eichler integrals of unary $\theta$ series is a vector-valued modular resurgent series. Concretely, its Borel transform has a tower of simple poles at $\rho_m = A m$ with $A = \sqrt{\pi i/(2N)}$, the resurgent series are constant Stokes vectors $\mathrm{St}_m^{(\nu)} = 2i(-1)^\nu m^{1-\nu} \Omega^{(\nu)}(S)\,\omega^{(\nu)}(m;N)$, and the Stokes–Dirichlet vector decomposes as a linear combination of Dirichlet $L$-functions (Proposition 4.7). Proposition 4.9 proves that $\mathrm{EI}_N^{(\nu)}$ is a vector-valued holomorphic quantum modular form of weight $3/2-\nu$, and Proposition 4.10 proves that median resummation at angle $\pi/2$ reconstructs the vector, up to the known pole term $i/(\pi\tau)$ in the $\nu=0$ case. The paradigm diagram closes because the same $S$-matrix appears in the functional equation of the completed Dirichlet vector: modularity of the $\theta$ series dictates the Stokes data.

Load-bearing premise

For the general claim of Section 4.6, the load-bearing premise is that the Dirichlet series of the Stokes constants are linear combinations of $L$-functions with Euler products (part 3 of Definition 2.5); the paper only proves this decomposition in the unary theta example and explicitly notes it is unavailable for arbitrary cusp forms.

Editorial extensions

If this is right

  • For every $N$ and $k$, the false theta function $\mathrm{EI}[\theta^{(1)}_{N,k}]$ can be recovered from its divergent asymptotic expansion by median resummation at angle $\pi/2$.
  • The $S$-matrix $\Omega^{(\nu)}(S)$ is encoded in the Stokes constants, so resurgent data of an Eichler integral directly reveals the modular transformation of the underlying theta series.
  • The vector-valued framework is essential precisely when the multiplier system is non-trivial; in the q-Pochhammer example, a Dirichlet-character twist diagonalizes the vector into scalar MRSs.
  • For any vector-valued cusp form of weight $1/2$ or $3/2$, the Eichler integral satisfies parts 1 and 2 of Definition 2.5 and is a vector-valued quantum modular form; at weight $1/2$ the Serre–Stark theorem reduces the claim to unary theta series, so the genuinely new examples occur at weight $3/2$.
  • The paradigm diagram closes without introducing an independent companion q-series; the second row is the $S$-image of the first, showing classical modularity is the organising principle.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Relaxing part 3 of Definition 2.5 to allow meromorphic continuation without Euler products would make the weight-$3/2$ cusp-form claim unconditional; the paper already proves all analytic aspects for general $g$, so the only missing input is arithmetic.
  • The one-sided sparse tower for general cusp forms suggests a natural extension of vector-valued resurgence to higher-depth or non-quadratic exponents, where the equally spaced tower must be abandoned.
  • The kernel mechanism suggests that modular anomalies of false theta functions are residue sweeps under contour rotation, which should generalise to higher-rank false theta functions and plumbed three-manifold invariants.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper introduces a vector-valued extension of the modular resurgence framework, defining vector-valued modular resurgent series (Definition 2.5), formulating a paradigm and two conjectures (median resummation and quantum modularity), and presenting two families of examples. For the q-Pochhammer vectors (Section 3), the vector-valued structure is explicitly identified as a repackaging of the scalar results of [2], with the Dirichlet series decomposed via Dirichlet characters. For Eichler integrals of unary theta series (Sections 4.1–4.5), the paper proves in detail that the folded vectors are vector-valued MRSs: it computes the Borel transform and Stokes constants (Proposition 4.7), shows the paradigm diagram (Eq. (173)), proves vector-valued quantum modularity (Proposition 4.9) and median resummation (Proposition 4.10), and discusses bimodular completions. The final subsection (Section 4.6) outlines an extension to arbitrary vector-valued cusp forms of weight 1/2 and 3/2, but with explicit caveats given in Remark 4.5. The central tension is that the abstract and conclusions assert the general weight-1/2 and 3/2 result without the qualification that part of Definition 2.5 is not verified in that generality.

Significance. The unary theta case is developed carefully and in detail, with explicit Borel transforms, Stokes constants rotated by the S-matrix, the closed paradigm diagram, and direct proofs of quantum modularity and median resummation. These computations are non-trivial and are a genuine contribution, as they provide a fully worked vector-valued resurgent structure for false theta functions. The q-Pochhammer section is honestly presented as a repackaging, and the Fourier-basis decomposition is clearly explained. The claimed general result for all weight-1/2 and 3/2 cusp forms, if proved, would be significant, but as written it exceeds what the manuscript establishes: Definition 2.5 requires an Euler-product L-function decomposition and a common two-sided equally spaced Borel tower, and Section 4.6 explicitly does not provide these. Thus the significance at the level of the unary theta example is solid, while the broader claim requires substantial qualification or additional proof.

major comments (4)
  1. [Abstract, Conclusions, and §4.6 (esp. Remark 4.5 and Eq. (228))] The statement that the modular resurgent structure holds 'in general' for Eichler integrals of any vector-valued cusp form of weight 1/2 or 3/2 is stronger than what is proved. Remark 4.5 states verbatim that part 3 of Definition 2.5 is not established by Steps 1–7, and Eq. (228) expresses the Stokes–Dirichlet vector only as a multiple of ϱ(S)LLL(s+κ−1), where LLL is not a vector of L-functions in the sense of Definition 2.3 because an Euler product would require g to be related to Hecke eigenforms. The abstract and Section 5 should either prove the Euler-product statement for a suitable class of eigenforms or explicitly restrict the 'general' claim to the weaker structure that is actually shown, with the unary theta case as the only fully verified vector-valued MRS in the strict sense of Definition 2.5.
  2. [§4.6, Step 3 (near Eq. (226))] Part 1 of Definition 2.5 requires, for every component, a common two-sided equally spaced tower of singularities {ρ_m = A m}_{m∈Z∖{0}}. For a general cusp form g, the singularities are {ξ_λ = 2πi λ} with λ ∈ α_k + Z_{≥0}, which are one-sided and depend on the component k; the authors themselves describe this as satisfying part 1 only in a 'weak sense' and note that the two-sided equally spaced tower in Section 4.3.2 relies on the unary folding ξ = ζ². Therefore the object studied in §4.6 does not, in general, satisfy Definition 2.5 as written. The definition or the claimed scope must be adjusted, for example by introducing a notion of weak or one-sided vector-valued MRS, or by removing the 'in general' assertion.
  3. [§4.6, Eq. (228) and Definition 2.5, part 3] Even putting aside the Euler product, the linear decomposition required in Eq. (38), namely L_±(s) = M_± diag(c_1,±^s, ..., c_{v,±}^s) L_±(s) with L_± a vector of genuine L-functions, is not demonstrated for general g. What Eq. (228) provides is a single scalar c = D and the vector LLL(s+κ−1) in place of a vector of L-functions, so the arithmetic part of the definition is not met. The paper should either fill this gap for Hecke eigenforms or state the result as a conditional resurgent structure that becomes an MRS only when the Dirichlet vector is known to decompose into L-functions.
  4. [§4.4.2 and §4.6, Step 7] The median resummation result for general g is proved directly in Step 7, bypassing the hypothesis of Conjecture 3; this is acceptable, but the wording in Step 7 that 'what Steps 1–6 establish is the conclusion of Conjecture 3' should be clarified to avoid implying that the hypothesis of Conjecture 3 has been verified. This is a presentation issue rather than a technical error, but it contributes to the overstatement of the general case.
minor comments (4)
  1. [§2.4.1 after Eq. (42)] There is a typo: 'witth Stokes vectors' should be 'with Stokes vectors'.
  2. [§4.3.2, Eq. (155) and surrounding text] The relationship between the Borel variable ζ and the variable ξ = ζ² is stated clearly, but the text would benefit from a remark that the two-sided tower in ζ is inherited by the folded vector only after the unary folding; this would help readers see why the general case in §4.6 is different.
  3. [§4.5, Eq. (202)] The notation in Eq. (202) uses τ both as a positive real parameter and, later, as the modular variable; this is a source of potential confusion and should be distinguished, e.g., by using t or a different letter for the positive real parameter.
  4. [§2.4.2, Conjectures 3 and 4] The conjectures are stated for vectors of q-series, but the examples in Section 4 require a subtraction of the singular term i/(πτ) for ν=0 (Eq. (190)); the conjecture statements could mention the possibility of such elementary subtractions for non-cuspidal inputs.

Circularity Check

1 steps flagged · score 2.0 of 10

No construction-level circularity: the Eichler-integral resurgent structure is derived from classical theta modularity, Hecke/Mellin correspondences and explicit Borel residues; the only inherited ingredient is the openly labelled q-Pochhammer repackaging from the authors' earlier work, and the §4.6 'general' claim is an overstatement of its own caveats rather than a circular reduction.

  1. self citation load bearing [Section 3.2.3, Lemma 3.5 and Lemma 3.6]
    "In this subsection, we show that the q-series vectors fN and gN satisfy Conjectures 3 and 4, as inherited component-wise from [2]. ... Lemma 3.5 ... Proof. The statement follows from [2, Eqs. (3.34) and (3.38)]. Lemma 3.6 ... Proof. The statement follows from [2, Thm. (3.10)]."

    The Section 3 verification of median resummation and vector-valued quantum modularity is not proved in the present paper; it is quoted from the same authors' earlier article [2]. The paper itself describes the vector framework for q-Pochhammer symbols as 'a convenient repackaging of scalar results', so this is a self-citation load-bearing for the illustrative Section 3 example. It is not load-bearing for the central Eichler-integral claim, which is derived from classical modularity of theta series and independent L-function decompositions.

full rationale

The Eichler-integral part of the paper is self-contained at the level of construction. Proposition 4.7 computes the Borel transform and Stokes vectors directly from the coefficient formula of Lemma 4.4; Lemma 4.3 gives the decomposition of the theta-Mellin Dirichlet series into Dirichlet L-functions; Lemma 4.8 derives the discontinuity from those computed residues; Proposition 4.9 proves quantum modularity through explicit period integrals; Proposition 4.10 proves median resummation by Borel–Laplace and Gamma-integral arguments. No fitted parameter is renamed as a prediction, and no derived quantity is defined in terms of the quantity it is supposed to predict. The only inherited step is Section 3, which openly credits [2] and is explicitly labelled a repackaging; it is a minor self-citation, not a circular main argument. Separately, the abstract and Conclusions claim the same modular resurgent structure for the Eichler integral of any vector-valued cusp form of weight 1/2 or 3/2, but Section 4.6, Step 3 admits that the Borel tower is one-sided and sparse for general g, and Remark 4.5 states verbatim that 'part 3 of Definition 2.5 is not established by Steps 1–7' because an Euler product 'would need g to be related to Hecke eigenforms'. That is an internal overstatement of the definition's requirements, not a circular reduction by construction, so it is weighed as a correctness risk rather than as circularity. Overall circularity score: 2.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters were fitted. The constants that appear, such as A = sqrt(pi i / (2N)) and the S-matrix entries, are derived from modularity and the definitions of the q-series, not chosen to match data. The only conceptual novelty, the definition of vector-valued modular resurgent series, is an organizing definition whose claims are tested by explicit examples rather than a new physical entity with independent evidence.

assumptions (6)
  • standard math Ecalle resurgence and Borel-Laplace summation theory, including median resummation identities.
    Used throughout Section 2.1 and in the proofs of Propositions 4.7 and 4.10; accepted background, not re-derived.
  • standard math Zagier's notion of quantum modular forms and the cocycle definition.
    Definition 2.2 and Proposition 4.9 rely on this framework.
  • domain assumption Classical vector-valued modularity of unary theta series, including the metaplectic multiplier and Poisson summation.
    Lemma 4.1 is the base of the Eichler-integral analysis and of the Hecke correspondence in Lemma 4.2.
  • domain assumption Serre-Stark theorem and Shimura's theory of half-integral weight modular forms.
    Used in Remark 4.5 to delimit the weight-1/2 case and in the general Section 4.6.
  • domain assumption Lawrence-Zagier lemma on L-series of periodic functions and Bernoulli-polynomial evaluations.
    Lemma 4.5 and Eq. (147) underpin the asymptotic expansions of Proposition 4.6.
  • domain assumption Vector-valued Hecke correspondence giving entireness and functional equation of the completed Dirichlet vector.
    Step 1 of Section 4.6; standard automorphic background, invoked without proof.

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Pith. "Pith review of Modular resurgent structures for vectors." pith.science (2026). https://pith.science/paper/XEOUDEQK

@misc{pith2026260808902,
  author       = {Pith},
  title        = {Pith review of: Modular resurgent structures for vectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XEOUDEQK}},
  note         = {Machine review of arXiv:2608.08902}
}
abstract

Building on prior results [1], we introduce vector-valued modular resurgent series, whose components exhibit a single infinite tower of singularities in the Borel plane, trivial secondary resurgent series, and Stokes constants given by linear combinations of the coefficients of a vector of $L$-functions. We extend the paradigm of modular resurgence to this setting, emphasizing the role of the Stokes constants and the interplay between the associated vectors of $q$-series and Dirichlet series, and describing the resulting symmetry relating canonical pairs of vector-valued modular resurgent series. Moreover, we conjecture that certain vectors of $q$-series with modular resurgent asymptotics are vector-valued quantum modular forms and can be reconstructed via median resummation. Finally, we show that vectors of $q$-Pochhammer symbols, previously considered in [2], and Eichler integrals of vector-valued modular forms of weight $1/2$ and $3/2$ can be studied within the framework of vector-valued modular resurgence. While the first case amounts to a convenient repackaging of scalar modular resurgence, the second involves a non-trivial representation of the modular group and therefore illustrates the necessity of the vector-valued framework; we establish its modular resurgent structure in general, and work it out in full detail for the unary theta series, whose Eichler integrals are the false theta functions of quantum topology.

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Works this paper leans on

36 extracted references · 24 canonical work pages

  1. [2]

    Modular resurgence, q-Pochhammer symbols, and quantum operators from mirror curves,

    V . Fantini and C. Rella, “Modular resurgence, q-Pochhammer symbols, and quantum operators from mirror curves,”Lett. Math. Phys.116no. 2, (2026) 39,arXiv:2506.08265 [hep-th]

    arXiv 2026

  2. [1]

    Modular resurgent structures,

    V . Fantini and C. Rella, “Modular resurgent structures,”arXiv:2404.11550 [math.NT]

    arXiv

  3. [3]

    ´Ecalle,Les fonctions r ´esurgentes

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

    work page 1981

  4. [4]

    Mitschi and D

    C. Mitschi and D. Sauzin,Divergent Series, Summability and Resurgence I, vol. 2153 ofLecture Notes in Mathematics. Springer, 2016

    work page 2016

  5. [5]

    Quantum modular forms,

    D. Zagier, “Quantum modular forms,”Quanta of Maths11(2010) 659–675

    work page 2010

  6. [6]

    HIM Lectures. Trimester Programm on Dynamics: Topology and Numbers

    D. Zagier, “HIM Lectures. Trimester Programm on Dynamics: Topology and Numbers.” https://www.youtube.com/watch?v=2Rj_xh3UKrU. 50

    work page

  7. [7]

    Strong-weak symmetry and quantum modularity of resurgent topological strings on localP 2,

    V . Fantini and C. Rella, “Strong-weak symmetry and quantum modularity of resurgent topological strings on localP 2,”Commun. Num. Theor. Phys.19no. 1, (2025) 1–73,arXiv:2404.10695 [hep-th]

    arXiv 2025

  8. [8]

    Eine verallgemeinerung der abelschen integrale,

    M. Eichler, “Eine verallgemeinerung der abelschen integrale,”Mathematische Zeitschrift67no. 1, (1957) 267–298

    work page 1957

Show all 36 references
  1. [9]

    Sur les int ´egrales attach´ees aux formes automorphes,

    G. Shimura, “Sur les int ´egrales attach´ees aux formes automorphes,”Journal of the Mathematical Society of Japan11no. 4, (1959) 291–311

    1959

  2. [10]

    Modular forms and quantum invariants of 3-manifolds,

    R. Lawrence and D. Zagier, “Modular forms and quantum invariants of 3-manifolds,”Asian Journal of Mathematics 31(1999) 93–108

    1999

  3. [11]

    On the quantum invariant for the Brieskorn homology spheres,

    K. Hikami, “On the quantum invariant for the Brieskorn homology spheres,”Int. J. Math.16 no. 06, (2005) 661–685,arXiv:0405028 [math-ph]

    2005

  4. [12]

    BPS spectra and 3-manifold invariants,

    S. Gukov, D. Pei, P. Putrov, and C. Vafa, “BPS spectra and 3-manifold invariants,”J. Knot Theor. Ramifications29no. 02, (2020) 2040003,arXiv:1701.06567 [hep-th]

    2020 arXiv

  5. [13]

    3d Modularity,

    M. C. N. Cheng, S. Chun, F. Ferrari, S. Gukov, and S. M. Harrison, “3d Modularity,”JHEP10 (2019) 010,arXiv:1809.10148 [hep-th]

    2019 arXiv

  6. [14]

    3d Modularity Revisited,

    M. C. N. Cheng, I. Coman, P. Kucharski, D. Passaro, and G. Sgroi, “3d Modularity Revisited,” arXiv:2403.14920 [hep-th]

  7. [15]

    Nahm sums, stability and the colored Jones polynomial,

    S. Garoufalidis and T. T. Q. Le, “Nahm sums, stability and the colored Jones polynomial,”Math. Sci.2no. 1, (2015) ,arXiv:1112.3905 [math.GT]

    2015 arXiv

  8. [16]

    q-series and tails of colored Jones polynomials,

    P. Beirne and R. Osburn, “q-series and tails of colored Jones polynomials,”Indag. Math.28no. 1, (2017) 247–260,arXiv:1601.05577 [math.NT]

    2017 arXiv

  9. [17]

    On the triplet vertex algebra W(p),

    D. Adamovic and A. Milas, “On the triplet vertex algebra W(p),”Adv. Math.217(2008) 2664–2699,arXiv:0707.1857 [math.QA]

    2008 arXiv

  10. [18]

    W-Algebras, False Theta Functions and Quantum Modular Forms, I,

    K. Bringmann and A. Milas, “W-Algebras, False Theta Functions and Quantum Modular Forms, I,”Int. Math. Res. Not.2015no. 21, (2015) 11351–11387

    2015

  11. [19]

    Half-integral weight eichler integrals and quantum modular forms,

    K. Bringmann and L. Rolen, “Half-integral weight eichler integrals and quantum modular forms,” Journal of Number Theory161(2016) 240–254. Special Issue on Applications of Automorphic Forms in Number Theory and Combinatorics

    2016

  12. [20]

    Quantum modularity of partial theta series with periodic coefficients,

    A. Goswami and R. Osburn, “Quantum modularity of partial theta series with periodic coefficients,”Forum Math.33no. 2, (2021) 451–463,arXiv:2012.02457 [math.NT]

    2021 arXiv

  13. [21]

    Going to the Other Side via the Resurgent Bridge,

    O. Costin, G. V . Dunne, A. Gruen, and S. Gukov, “Going to the Other Side via the Resurgent Bridge,”arXiv:2310.12317 [hep-th]

    arXiv

  14. [22]

    On Uniqueness of Mock Theta Functions,

    O. Costin, G. V . Dunne, and A. Saraeb, “On Uniqueness of Mock Theta Functions,” arXiv:2604.20034 [math.NT]

    arXiv

  15. [23]

    Orientation reversal and the Chern-Simons natural boundary,

    G. Adams, O. Costin, G. V . Dunne, S. Gukov, and O. ¨Oner, “Orientation reversal and the Chern-Simons natural boundary,”JHEP08(2025) 154,arXiv:2505.14441 [hep-th]

    2025 arXiv

  16. [24]

    M. C. N. Cheng, I. Coman, and M. Rossell ´o.Work in progress. 51

  17. [25]

    M. C. N. Cheng, I. Coman, and M. Rossell ´o.Work in progress

  18. [26]

    A framework for modular properties of false theta functions,

    K. Bringmann and C. Nazaroglu, “A framework for modular properties of false theta functions,” Res. Math. Sci.6no. 3, (2019) 30,arXiv:1904.05377 [math.NT]

    2019 arXiv

  19. [27]

    An introduction to resurgence in quantum theory

    M. Mari ˜no, “An introduction to resurgence in quantum theory.”https://www. marcosmarino.net/uploads/1/3/3/5/133535336/resurgence-course.pdf. Lecture notes

  20. [28]

    Resurgence, Stokes constants, and arithmetic functions in topological string theory,

    C. Rella, “Resurgence, Stokes constants, and arithmetic functions in topological string theory,” Commun. Num. Theor. Phys.17no. 3, (2023) 709–820,arXiv:2212.10606 [hep-th]

    2023 arXiv

  21. [29]

    Resurgent Lambert series with characters,

    D. Broadhurst and D. Dorigoni, “Resurgent Lambert series with characters,” arXiv:2507.21352 [math.NT]

    arXiv

  22. [30]

    Rademacher Sums and Rademacher Series,

    M. C. N. Cheng and J. F. R. Duncan, “Rademacher Sums and Rademacher Series,”Contrib. Math. Comput. Sci.8(2014) 143–182,arXiv:1210.3066 [math.NT]

    2014 arXiv

  23. [31]

    Resurgence in complex Chern-Simons theory,

    S. Gukov, M. Marino, and P. Putrov, “Resurgence in complex Chern-Simons theory,” arXiv:1605.07615 [hep-th]

    arXiv

  24. [32]

    Quantum Modular Forms and Resurgence,

    E. McSpirit and L. Rolen, “Quantum Modular Forms and Resurgence,”arXiv:2505.00799 [math.NT]

    arXiv

  25. [33]

    T. M. Apostol,Introduction to Analytic Number Theory. Springer-Verlag, New York, 1976

    1976

  26. [34]

    On modular forms of half integral weight,

    G. Shimura, “On modular forms of half integral weight,”Annals of Mathematics97no. 3, (1973) 440–481

    1973

  27. [35]

    Modular forms of weight1/2,

    J.-P. Serre and H. M. Stark, “Modular forms of weight1/2,” inModular Functions of One Variable VI, vol. 627 ofLecture Notes in Mathematics, pp. 27–67. Springer, Berlin, Heidelberg, 1977

    1977

  28. [36]

    Zwegers,Mock Theta Functions

    S. Zwegers,Mock Theta Functions. Ph.D. Thesis, Utrecht University, 2002. 52

    2002

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