Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 101, 69 pages      arXiv:2407.01412      https://doi.org/10.3842/SIGMA.2025.101

The Regularity of ODEs and Thimble Integrals with Respect to Borel Summation

Veronica Fantini a and Aaron Fenyes b
a) Laboratoire Mathématique d'Orsay, France
b) Studio Infinity, USA

Received January 06, 2025, in final form November 05, 2025; Published online December 03, 2025

Abstract
Through Borel summation, one can often reconstruct an analytic solution of a problem from its asymptotic expansion. We view the effectiveness of Borel summation as a regularity property of the solution, and we show that the solutions of certain differential equation and integration problems are regular in this sense. By taking a geometric perspective on the Laplace and Borel transforms, we also clarify why ''Borel regular'' solutions are associated with special points on the Borel plane. The particular classes of problems we look at are level $1$ ODEs and exponential period integrals over one-dimensional Lefschetz thimbles. To expand the variety of examples available in the literature, we treat various examples of these problems in detail.

Key words: Borel summation; Laplace transform; Borel transform; resurgence; translation surfaces; thimble integrals; ordinary differential equations; irregular singularities; divergent series; asymptotics; Stokes phenomena; Stokes constants; Airy function; Airy-Lucas function; generalized Airy function; modified Bessel equation; triangular cantilever.

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References

  1. Adam J.A., The mathematical physics of rainbows and glories, Phys. Rep. 356 (2002), 229-365.
  2. Andersen J.E., Resurgence analysis of meromorphic transforms, arXiv:2011.01110.
  3. Aniceto I., Başar G., Schiappa R., A primer on resurgent transseries and their asymptotics, Phys. Rep. 809 (2019), 1-135, arXiv:1802.10441.
  4. Argyres P.C., Ünsal M., The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects, J. High Energy Phys. 2012 (2012), no. 8, 063, 98 pages, arXiv:1206.1890.
  5. Arnold V.I., Gusein-Zade S.M., Varchenko A.N., Singularities of differentiable maps. Vol. II. Monodromy and asymptotic integrals, Monogr. Math., Vol. 83, Birkhäuser, Boston, MA, 1988.
  6. Baldino S., Schiappa R., Schwick M., Vega R., Resurgent Stokes data for Painlevé equations and two-dimensional quantum (super) gravity, Commun. Number Theory Phys. 17 (2023), 385-552, arXiv:2203.13726.
  7. Balser W., Formal power series and linear systems of meromorphic ordinary differential equations, Universitext, Springer, New York, 2000.
  8. Barbieri A., Bridgeland T., Stoppa J., A quantized Riemann-Hilbert problem in Donaldson-Thomas theory, Int. Math. Res. Not. 2022 (2022), 3417-3456, arXiv:1905.00748.
  9. Becchetti F.D., The nuclear optical model and its optical-scattering analog: Mie scattering, American J. Phys. 91 (2023), 637-643.
  10. Behtash A., Dunne G.V., Schäfer T., Sulejmanpasic T., Ünsal M., Toward Picard-Lefschetz theory of path integrals, complex saddles and resurgence, Ann. Math. Sci. Appl. 2 (2017), 95-212, arXiv:1510.03435.
  11. Beilinson A.A., Drinfeld V.G., Quantization of Hitchin's fibration and Langlands' program, in Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), Math. Phys. Stud., Vol. 19, Kluwer Academic Publishers, Dordrecht, 1996, 3-7.
  12. Berry M., Asymptotics, superasymptotics, hyperasymptotics$\,\dots$, in Asymptotics Beyond All Orders (La Jolla, CA, 1991), NATO Adv. Sci. Inst. Ser. B: Phys., Vol. 284, Plenum, New York, 1991, 1-14.
  13. Berry M., Howls C., Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. London Ser. A 434 (1991), 657-675.
  14. Borel E., Mémoire sur les séries divergentes, Ann. Sci. École Norm. Sup. (3) 16 (1899), 9-131.
  15. Borel E., Leçons sur les séries divergentes, Monatsh. Math. 13 (1902), 8-8.
  16. Braaksma B.L.J., Laplace integrals in singular differential and difference equations, in Ordinary and Partial Differential Equations, Lecture Notes in Math., Vol. 827, Springer, Berlin, 1980, 25-53.
  17. Bridgeland T., Riemann-Hilbert problems from Donaldson-Thomas theory, Invent. Math. 216 (2019), 69-124, arXiv:1611.03697.
  18. Bridgeland T., Riemann-Hilbert problems for the resolved conifold, J. Differential Geom. 115 (2020), 395-435, arXiv:1703.02776.
  19. Cecotti S., Vafa C., On classification of $N=2$ supersymmetric theories, Comm. Math. Phys. 158 (1993), 569-644, arXiv:hep-th/9211097.
  20. Charbonnier S., Chidambaram N.K., Garcia-Failde E., Giacchetto A., Shifted Witten classes and topological recursion, Trans. Amer. Math. Soc. 377 (2024), 1069-1110, arXiv:2203.16523.
  21. Chiani M., Dardari D., Simon M.K., New exponential bounds and approximations for the computation of error probability in fading channels, IEEE Trans. Wireless Commun. 2 (2003), 840-845.
  22. Costin O., On Borel summation and Stokes phenomena for rank-$1$ nonlinear systems of ordinary differential equations, Duke Math. J. 93 (1998), 289-344, arXiv:math.CA/0608408.
  23. Costin O., Topological construction of transseries and introduction to generalized Borel summability, in Analyzable Functions and Applications, Contemp. Math., Vol. 373, American Mathematical Society, Providence, RI, 2005, 137-175, arXiv:math.CA/0608309.
  24. Costin O., Asymptotics and Borel summability, Chapman & Hall/CRC, Vol. 141, CRC Press, Boca Raton, FL, 2009.
  25. Costin O., Dunne G.V., Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I, J. Phys. A 52 (2019), 445205, 29 pages, arXiv:1904.11593.
  26. Costin O., Kruskal M.D., On optimal truncation of divergent series solutions of nonlinear differential systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), 1931-1956, arXiv:math.CA/0608410.
  27. Delabaere E., Divergent series, summability and resurgence. III. Resurgent methods and the first Painlevé equation, Lecture Notes in Math., Vol. 2155, Springer, Cham, 2016.
  28. Delabaere E., Dillinger H., Pham F., Exact semiclassical expansions for one-dimensional quantum oscillators, J. Math. Phys. 38 (1997), 6126-6184.
  29. Delabaere E., Howls C., Global asymptotics for multiple integrals with boundaries, Duke Math. J. 112 (2002), 199-264.
  30. Delabaere E., Pham F., Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), 1-94.
  31. Delabaere E., Rasoamanana J.-M., Sommation effective d'une somme de Borel par séries de factorielles, Ann. Inst. Fourier (Grenoble) 57 (2007), 421-456, arXiv:math.CV/0602237.
  32. Deligne P., Malgrange B., Ramis J.-P., Singularités irrégulières, Doc. Math. (Paris), Vol. 5, Société Mathématique de France, Paris, 2007.
  33. der Hoeven J.V., Complex transseries solutions to algebraic differential equations, 2008, https://www.texmacs.org/joris/osc/osc-abs.html.
  34. Dingle R.B., Asymptotic expansions: their derivation and interpretation,Academic Press, London, 1973.
  35. Dorigoni D., An introduction to resurgence, trans-series and alien calculus, Ann. Physics 409 (2019), 167914, 38 pages, arXiv:1411.3585.
  36. Drazin P.G., Reid W.H., Hydrodynamic stability, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge, 2004.
  37. Dubrovin B., Geometry and integrability of topological-antitopological fusion, Comm. Math. Phys. 152 (1993), 539-564, arXiv:hep-th/9206037.
  38. Dubrovin B., Kapaev A., A Riemann-Hilbert approach to the Heun equation, SIGMA 14 (2018), 093, 24 pages, arXiv:1809.02311.
  39. Dunne G.V., Ünsal M., What is QFT? Resurgent trans-series, Lefschetz thimbles, and new exact saddles, Proc. Sci. 251 (2016), PoS(LATTICE 2015)010, 20 pages, arXiv:1511.05977.
  40. Écalle J., Les fonctions résurgentes. Tome I, Publ. Math. Orsay, Vol. 5, Université de Paris-Sud, Département de Mathématiques, Orsay, 1981.
  41. Écalle J., Les fonctions résurgentes. Tome II, Publ. Math. Orsay, Vol. 6, Université de Paris-Sud, Département de Mathématiques, Orsay, 1981.
  42. Écalle J., Les fonctions résurgentes. Tome III, Publ. Math. Orsay, Vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985.
  43. Écalle J., Six lectures on transseries, analysable functions and the constructive proof of Dulac's conjecture, in Bifurcations and Periodic Orbits of Vector Fields (Montreal, PQ, 1992), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., Vol. 408, Kluwer Academic Publishers, Dordrecht, 1993, 75-184.
  44. Elbaek M.W., Quantum invariants and Chern-Simons theory, Ph.D. Thesis, Centre for Quantum Geometry of Moduli Spaces (QGM), Aarhus University, 2019.
  45. Fantini V., Fenyes A., Regular singular Volterra equations on complex domains, SIAM J. Math. Anal. 57 (2025), 336-363, arXiv:2309.00603.
  46. Fauvet F., Menous F., Quéva J., Resurgence and holonomy of the $\phi^{2k}$ model in zero dimension, J. Math. Phys. 61 (2020), 092301, 23 pages, arXiv:1910.01606.
  47. Fenyes A., IHES lectures, 2023, https://www.youtube.com/watch?v=L11uTATilDo.
  48. Fresàn J., Jossen P., Exponential motives, http://javier.fresan.perso.math.cnrs.fr/expmot.pdf.
  49. Gaiotto D., Moore G.W., Neitzke A., Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299 (2010), 163-224, arXiv:0807.4723.
  50. Garoufalidis S., Chern-Simons theory, analytic continuation and arithmetic, Acta Math. Vietnam. 33 (2008), 335-362, arXiv:0711.1716.
  51. Garoufalidis S., Gu J., Mariño M., Peacock patterns and resurgence in complex Chern-Simons theory, Res. Math. Sci. 10 (2023), 29, 67 pages, arXiv:2012.00062.
  52. Garoufalidis S., Lê T.T.Q., Mariño M., Analyticity of the free energy of a closed 3-manifold, SIGMA 4 (2008), 080, 20 pages, arXiv:0809.2572.
  53. Genta G., Vibration dynamics and control, Mech. Engrg. Ser., Springer, New York, 2009.
  54. Gukov S., Mariño M., Putrov P., Resurgence in complex Chern-Simons theory, arXiv:1605.07615.
  55. Gupta S., Meromorphic quadratic differentials with half-plane structures, Ann. Acad. Sci. Fenn. Math. 39 (2014), 305-347, arXiv:1301.0332.
  56. Hodgson P.E., The nuclear optical model, Rep. Progr. Phys. 34 (1971), 765-819.
  57. Howls C.J., Hyperasymptotics for integrals with finite endpoints, Proc. Roy. Soc. London Ser. A 439 (1992), 373-396.
  58. Howls C.J., Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem, Proc. Roy. Soc. London Ser. A 453 (1997), 2271-2294.
  59. Joshi K.D., Introduction to general topology, Halsted Press Book, John Wiley & Sons, Inc., New York, 1983.
  60. Kawai T., Takei Y., Algebraic analysis of singular perturbation theory, Transl. Math. Monogr., Vol. 227, American Mathematical Society, Providence, RI, 2005.
  61. Keen L., Lakic N., Hyperbolic geometry from a local viewpoint, London Math. Soc. Stud. Texts, Vol. 68, Cambridge University Press, Cambridge, 2007.
  62. Knoll J., Schaeffer R., Semiclassical scattering theory with complex trajectories. I. Elastic waves, Ann. Physics 97 (1976), 307-366.
  63. Kontsevich M., IHES lectures: exponential integrals, Part 1, 2015, https://www.youtube.com/watch?v=tM25X6AI5dY.
  64. Kontsevich M., Exponential integrals, Lefschetz thimbles and linear resurgence, ERC ReNewQuantum talk, 2020, https://www.youtube.com/watch?v=krf0pXjTl9s.
  65. Kontsevich M., Odesskii A., When the Fourier transform is one loop exact?, Selecta Math. (N.S.) 30 (2024), 35, 55 pages, arXiv:2306.02178.
  66. Kontsevich M., Soibelman Y., Analyticity and resurgence in wall-crossing formulas, Lett. Math. Phys. 112 (2022), 32, 56 pages.
  67. Kontsevich M., Soibelman Y., Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions, arXiv:2402.07343.
  68. Loday-Richaud M., Stokes phenomenon, multisummability and differential Galois groups, Ann. Inst. Fourier (Grenoble) 44 (1994), 849-906.
  69. Loday-Richaud M., Divergent series, summability and resurgence. II. Simple and multiple summability, Lecture Notes in Math., Vol. 2154, Springer, Cham, 2016.
  70. Loday-Richaud M., Remy P., Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems, J. Differential Equations 250 (2011), 1591-1630, arXiv:1006.2613.
  71. Malgrange B., Méthode de la phase stationnaire et sommation de Borel, in Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Phys., Vol. 126, Springer, Berlin, 1980, 170-177.
  72. Malgrange B., Fourier transform and differential equations, in Recent Developments in Quantum Mechanics, Math. Phys. Stud., Vol. 12, Kluwer Academic Publishers, Dordrecht, 1991, 33-48.
  73. Malgrange B., Sommation des séries divergentes, Exposition. Math. 13 (1995), 163-222.
  74. Malgrange B., Ramis J.-P., Fonctions multisommables, Ann. Inst. Fourier (Grenoble) 42 (1992), 353-368.
  75. Mariño M., Lectures on non-perturbative effects in large $N$ gauge theories, matrix models and strings, Fortschr. Phys. 62 (2014), 455-540, arXiv:1206.6272.
  76. Menous F., Les bonnes moyennes uniformisantes et une application à la resommation réelle, Ph.D. Thesis, Universitè P. Sabatier, 1999.
  77. Menous F., Novelli J.-C., Thibon J.-Y., Mould calculus, polyhedral cones, and characters of combinatorial Hopf algebras, Adv. in Appl. Math. 51 (2013), 177-227, arXiv:1109.1634.
  78. Miller P.D., Applied asymptotic analysis, Grad. Stud. Math., Vol. 75, American Mathematical Society, Providence, RI, 2006.
  79. Mitschi C., Sauzin D., Divergent series, summability and resurgence. I. Monodromy and resurgence, Lecture Notes in Math., Vol. 2153, Springer, Cham, 2016.
  80. Mladenov V., Mastorakis N. (Editors), Advanced topics on applications of fractional calculus on control problems, system stability and modeling, World Scientific and Engineering Academy and Society, 2014.
  81. Nevanlinna F., Zur Theorie der asymptotischen Potenzreihen I. II, Suomalaisen tiedeakademian kustantama, 1918.
  82. Nikolaev N., Exact solutions for the singularly perturbed Riccati equation and exact WKB analysis, Nagoya Math. J. 250 (2023), 434-469, arXiv:2008.06492.
  83. Nikolaev N., Existence and uniqueness of exact WKB solutions for second-order singularly perturbed linear ODEs, Comm. Math. Phys. 400 (2023), 463-517, arXiv:2106.10248.
  84. Olver F.W.J., Daalhuis O.A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V., Cohl H.S., McClain M.A. (Editors), NIST digital library of mathematical functions, Release 1.2.1 of 2024-06-15, https://dlmf.nist.gov/.
  85. Pham F., Vanishing homologies and the $n$ variable saddlepoint method, in Singularities, Part 2, Proc. Sympos. Pure Math., Vol. 40, American Mathematical Society, Providence, RI, 1983, 319-333.
  86. Pham F., La descente des cols par les onglets de Lefschetz, avec vues sur Gauss-Manin, Astérisque 130 (1985), 11-47.
  87. Pham F., Resurgence, quantized canonical transformations, and multi-instanton expansions, in Algebraic Analysis, Vol. II, Academic Press, Boston, MA, 1988, 699-726.
  88. Podlubny I., Correction: ''Geometric and physical interpretation of fractional integration and fractional differentiation'', Fract. Calc. Appl. Anal. 6 (2003), 109-110, arXiv:math.CA/0110241.
  89. Poincaré H., Sur les intégrales irrégulières, Acta Math. 8 (1886), 295-344.
  90. Ramis J.-P., Théorèmes d'indices Gevrey pour les équations différentielles ordinaires, Mem. Amer. Math. Soc. 48 (1984), viii+95 pages.
  91. Ramis J.-P., Séries divergentes et théories asymptotiques, Bull. Soc. Math. France 121 (1993), 1-74.
  92. Reid W.H., Composite approximations to the solutions of the Orr-Sommerfeld equation, Stud. Appl. Math. 51 (1972), 341-368.
  93. Sauzin D., Variations on the resurgence of the gamma function, arXiv:2112.15226.
  94. Schiff J.L., The Laplace transform. Theory and applications, Undergrad. Texts Math., Springer, New York, 1999.
  95. Sokal A.D., An improvement of Watson's theorem on Borel summability, J. Math. Phys. 21 (1980), 261-263.
  96. Sternin B.Yu., Shatalov V.E., Borel-Laplace transform and asymptotic theory. Introduction to resurgent analysis, CRC Press, Boca Raton, FL, 1996.
  97. Tanizaki Y., Lefschetz-thimble techniques for path integral of zero-dimensional $O(n)$ sigma models, Phys. Rev. D 91 (2015), 036002, 9 pages, arXiv:1412.1891.
  98. Voros A., The return of the quartic oscillator: the complex WKB method, Ann. Inst. H. Poincaré Sect. A (N.S.) 39 (1983), 211-338.
  99. Watson G.S., A theory of asymptotic series, Philos. Trans. Roy. Soc. A 211 (1912), 279-313.
  100. Witten E., Analytic continuation of Chern-Simons theory, in Chern-Simons Gauge Theory: 20 Years After, AMS/IP Stud. Adv. Math., Vol. 50, American Mathematical Society, Providence, RI, 2011, 347-446, arXiv:1001.2933.
  101. Zorich A., Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, 437-583, arXiv:math.DS/0609392.

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