Skip to main content
SpringerLink
Log in

Asymptotics, ambiguities and resurgence

  • Conference paper
Resurgence, Physics and Numbers

Part of the book series: CRM Series ((CRMSNS,volume 20))

Abstract

The appearance of resurgent functions in the context of the perturbative study of observables in physics is now well established. Whether these arise from the related study of non-linear systems or the saddle-point perturbative analysis, one is left with an asymptotic series and the need of a non-perturbative completion, or transseries, which includes different non-perturbative phenomena. The complete understanding of resummation procedures and the resurgence of the non-perturbative phenomena can then lead to a systematic approach to obtain exact results such as strong-weak coupling interpolation, cancellation of ambiguities in the so-called Stokes directions, and more generally the study of analytic properties of the respective transseries solutions. These notes will give a general overview of how to set-up resurgence in simple examples, and how to proceed towards exact analytic results.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Écalle, “Les fonctions résurgentes”, Vol. 1, Algeèbres de fonctions résurgentes, Publ. Math. Orsay, 81-05, 1981, 248 pp.

    Google Scholar 

  2. J. Écalle, “Les fonctions résurgentes”, Vol. 2, Les fonctions résurgentes appliquées à l’itératio, Publ. Math. Orsay, 81-06, 1981, 283 pp.

    Google Scholar 

  3. J. Écalle, “Les fonctions résurgentes”, Vol. 3, L’équation du pont et la classification analytique des objets locaux, Publ. Math. Orsay, 85-05, 1985, 585 pp.

    Google Scholar 

  4. B. Candelpergher, J. Nosmas and F. Pham, Premiers pas en calcul étranger, Ann. Inst. Fourier 43 (1993), 201.

    Article  MathSciNet  MATH  Google Scholar 

  5. O. Costin, Exponential asymptotics, transseries, and generalized Borel summation for analytic, nonlinear, rank-one systems of ordinary differential equations, Internat. Math. Res. Notices 8 (1995), 377. [arXiv:math.CA/0608414]

    Article  MathSciNet  MATH  Google Scholar 

  6. O. Costin, 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]

    Article  MathSciNet  MATH  Google Scholar 

  7. J. P. Boyd, The Devil’s invention: asymptotic, superasymptotic and hyperasymptotic series, Acta Appl. Math. 56 (1999), 1.

    Article  MathSciNet  MATH  Google Scholar 

  8. E. Delabaere and F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. Henri Poincaré 71 (1999), 1.

    MathSciNet  MATH  Google Scholar 

  9. T. Seara and D. Sauzin, Resumació de Borel i teoria de la ressurgència, Butl. Soc. Catalana Mat. 18 (2003), 131.

    MathSciNet  Google Scholar 

  10. D. Sauzin, Resurgent functions and splitting problems, RIMS Kokyuroku 1493 (2006), 48–117. [arXiv:0706.0137]

    Google Scholar 

  11. G. A. Edgar, Transseries for beginners, Real Anal. Exchange 35 (2009), 253. [arXiv:0801.4877]

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Mariño, Lectures on non-perturbative effects in large N Gauge theories, matrix models and mtrings, Fortsch. Phys. 62 (2014), 455–540. [arXiv:1206.6272]

    Article  MATH  Google Scholar 

  13. D. Sauzin, Introduction to 1-summability and Resurgence, In: “Divergent Series, Summability and Resurgence I, Monodromy and Resurgence, Part II”, Lecture Notes in Mathematics, Vol. 2153, Springer, Heidelberg, 2016, 121–293. [arXiv:1405.0356]

    Google Scholar 

  14. G. V. Dunne and M. Ünsal, What is QFT? Resurgent transseries, Lefschetz thimbles, and new exact saddles, In: “Proceedings, 33rd International Symposium on Lattice Field Theory (Lattice 2015)”, 2015. [arXiv:1511.05977]

    Google Scholar 

  15. M. Mariño, “Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory”, Cambridge University Press, 2015.

    Google Scholar 

  16. I. Aniceto, G. Başar and R. Schiappa, A primer on resurgent transseries and their asymptotics, upcoming (2017).

    Google Scholar 

  17. A. Olde Daalhuis, Hyperasymptotics for nonlinear ODEs I. A Riccati equation, Proceedings of the Royal Society of London A461 (2005), 2503–2520.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Olde Daalhuis, Hyperasymptotics for nonlinear ODEs II. The first Painlevé equation and a second-order Riccati equation, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences A461 (2005), no. 2062, 3005–3021.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Garoufalidis, A. Its, A. Kapaev and M. Mariño, Asymptotics of the instantons of Painlevé I, Int. Math. Res. Notices 2012 (2012), 561. [arXiv:1002.3634]

    Article  MATH  Google Scholar 

  20. I. Aniceto, R. Schiappa and M. Vonk, The resurgence of instantons in string theory, Commun. Num. Theor. Phys. 6 (2012), 339. [arXiv:1106.5922]

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Schiappa and R. Vaz, The resurgence of instantons: multi-cut Stokes phases and the Painlevé II equation, Commun. Math. Phys. 330 (2014), 655–721. [arXiv:1302.5138]

    Article  MATH  Google Scholar 

  22. O. Costin, R. D. Costin and M. Huang, A direct method to find Stokes multipliers in closed form for P1 and more general integrable systems, Trans. Amer. Math. Soc. (2012). [arXiv:1205.0775]

    Google Scholar 

  23. C. M. Bender and T. T. Wu, Anharmonic oscillator, Phys. Rev. 184 (1969), 1231.

    Article  MathSciNet  Google Scholar 

  24. C. M. Bender and T. Wu, Anharmonic oscillator 2: a study of perturbation theory in large order, Phys. Rev. D7 (1973), 1620.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Zinn-Justin, Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation, Phys. Rept. 70 (1981), 109.

    Article  MathSciNet  Google Scholar 

  27. M. Beneke, Renormalons, Phys. Rept. 317 (1999), 1. [arXiv:hep-ph/9807443]

    Article  Google Scholar 

  28. E. Bogomolny, Calculation of instanton—anti-instanton contributions in quantum mechanics, Phys. Lett. B91 (1980), 431.

    Article  Google Scholar 

  29. J. Zinn-Justin, Multi-instanton contributions in quantum mechanics, Nucl. Phys. B192 (1981), 125–140.

    Article  MathSciNet  Google Scholar 

  30. J. Zinn-Justin, Multi-instanton contributions in quantum mechanics. 2, Nucl. Phys. B218 (1983), 333–348. http://dx.doi.org/10.1016/0550-3213(83)90369-3

    Article  MathSciNet  Google Scholar 

  31. J. Zinn-Justin, From multi-instantons to exact results, Ann. Inst. Fourier 53 (2003) 1259.

    Article  MathSciNet  MATH  Google Scholar 

  32. J. Zinn-Justin and U. D. Jentschura, Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions, Annals Phys. 313 (2004), 197. http://arXiv.org/abs/quant-ph/0501136arXiv:quant-ph/0501136

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Zinn-Justin and U. D. Jentschura, Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations, Annals Phys. 313 (2004), 269. [arXiv:quant-ph/0501137]

    Article  MathSciNet  MATH  Google Scholar 

  34. U. D. Jentschura and J. Zinn-Justin, Instantons in quantum mechanics and resurgent expansions, Phys. Lett. B596 (2004), 138. [arXiv:hep-ph/0405279]

    Article  MathSciNet  MATH  Google Scholar 

  35. U. D. Jentschura, A. Surzhykov and J. Zinn-Justin, Multi-instantons and exact results. III: unification of even and odd anharmonic oscillators, Annals Phys. 325 (2010), 1135–1172.

    Article  MathSciNet  MATH  Google Scholar 

  36. U. D. Jentschura and J. Zinn-Justin, Multi-instantons and exact results. IV: path integral formalism, Annals Phys. 326 (2011) 2186–2242.

    Article  MathSciNet  MATH  Google Scholar 

  37. G. V. Dunne and M. Ünsal, Generating nonperturbative physics from perturbation theory, Phys. Rev. D89 (2014), no. 4, 041701. [arXiv:1306.4405]

    Google Scholar 

  38. G. Başar, G. V. Dunne and M. Ünsal, Resurgence Theory, Ghost-instantons, and Analytic Continuation of Path Integrals, JHEP 10 (2013), 041. [arXiv:1308.1108]

    MathSciNet  MATH  Google Scholar 

  39. I. Aniceto and R. Schiappa, Nonperturbative ambiguities and the reality of resurgent transseries, Commun. Math. Phys. 335 (2015), no. 1, 183–245.[arXiv:1308.1115]

    Article  MathSciNet  MATH  Google Scholar 

  40. G. V. Dunne and M. Ünsal, Uniform WKB, Multi-Instantons, and Resurgent Trans-Series, Phys. Rev. D89 (2014), no. 10, 105009. [arXiv:1401.5202]

    Google Scholar 

  41. G. Başar and G. V. Dunne, Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems, JHEP 1502 (2015), 160. [arXiv:1501.05671]

    Google Scholar 

  42. T. Misumi, M. Nitta and N. Sakai, Resurgence in sine-Gordon quantum mechanics: exact agreement between multi-instantons and uniform WKB, JHEP 09 (2015), 157. [arXiv:1507.00408]

    Article  MathSciNet  Google Scholar 

  43. F. David, Phases of the large N matrix model and nonperturbative effects in 2-d gravity, Nucl. Phys. B348 (1991), 507–524.

    Article  MathSciNet  Google Scholar 

  44. F. David, Nonperturbative effects in matrix models and vacua of two-dimensional gravity, Phys. Lett. B302 (1993), 403–410. [arXiv:hep-th/9212106]

    Article  MathSciNet  Google Scholar 

  45. M. Mariño, R. Schiappa and M. Weiss, Nonperturbative effects and the large-order behavior of matrix models and topological strings, Commun. Num. Theor. Phys. 2 (2008), 349. [arXiv:0711.1954]

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 0812 (2008), 114. [arXiv:0805.3033]

    Article  MathSciNet  MATH  Google Scholar 

  47. M. Mariño, R. Schiappa and M. Weiss, Multi-instantons and multi-cuts, J. Math. Phys. 50 (2009), 052301. [arXiv:0809.2619]

    Article  MathSciNet  MATH  Google Scholar 

  48. S. Pasquetti and R. Schiappa, Borel and Stokes nonperturbative phenomena in topological string theory and c = 1 matrix models, Annales Henri Poincaré 11 (2010), 351. [arXiv:0907.4082]

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Mariño, S. Pasquetti and P. Putrov, Large N duality beyond the genus expansion, JHEP 07 (2010), 074. [arXiv:0911.4692]

    Article  MathSciNet  MATH  Google Scholar 

  50. J. G. Russo, A note on perturbation series in supersymmetric gauge theories, JHEP 1206 (2012), 038. [arXiv:1203.5061]

    Article  MathSciNet  Google Scholar 

  51. I. Aniceto, J. G. Russo and R. Schiappa, Resurgent analysis of localizable observables in supersymmetric gauge theories, JHEP 1503 (2015), 172. [arXiv:1410.5834]

    Article  MathSciNet  Google Scholar 

  52. R. Couso-Santamaría, R. Schiappa and R. Vaz, Finite N from resurgent large N, Annals Phys. 356 (2015), 1–28. [arXiv:1501.01007]

    Article  MathSciNet  MATH  Google Scholar 

  53. M. P. Heller and M. Spaliński, Hydrodynamics beyond the gradient expansion: resurgence and resummation, Phys. Rev. Lett. 115 (2015), no. 7, 072501. [arXiv:1503.07514]

    Article  Google Scholar 

  54. I. Aniceto, The Resurgence of the cusp anomalous dimension, J. Phys. A49 (2016), 065403. [arXiv:1506.03388]

    MathSciNet  MATH  Google Scholar 

  55. D. Dorigoni and Y. Hatsuda, Resurgence of the cusp anomalous dimension, JHEP 09 (2015), 138. [arXiv:1506.03763]

    Article  MathSciNet  Google Scholar 

  56. I. Aniceto and M. Spaliński, Resurgence in extended hydrodynamics, Phys. Rev. D93 (2016), 085008. [arXiv:1511.06358]

    MathSciNet  Google Scholar 

  57. P. Argyres and M. Ünsal, A semiclassical realization of infrared renormalons, Phys. Rev. Lett. 109 (2012), 121601. [arXiv:1204.1661]

    Article  Google Scholar 

  58. P. C. Argyres and M. Ünsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects, JHEP 1208 (2012), 063. [arXiv:1206.1890]

    Article  MathSciNet  Google Scholar 

  59. G. V. Dunne and M. Ünsal, Resurgence and trans-series in quantum field theory: the ℂ℉N−1 model, JHEP 1211 (2012), 170. [arXiv:1210.2423]

    Article  MathSciNet  Google Scholar 

  60. G. V. Dunne and M. Ünsal, Continuity and resurgence: towards a continuum definition of the ℂ℉N−1 model, Phys. Rev. D87 (2013), 025015. [arXiv:1210.3646]

    Google Scholar 

  61. A. Cherman, D. Dorigoni, G. V. Dunne and M. Ünsal, Resurgence in quantum field theory: nonperturbative effects in the principal chiral model, Phys. Rev. Lett. 112 (2014), 021601. [arXiv:1308.0127]

    Article  Google Scholar 

  62. A. Cherman, P. Koroteev and M. Ünsal, Resurgence and holomorphy: from weak to strong coupling, J. Math. Phys. 56 (2015), no. 5, 053505. [arXiv:1410.0388]

    Article  MathSciNet  MATH  Google Scholar 

  63. M. P. Bellon and P. J. Clavier, A Schwinger-Dyson equation in the Borel plane: singularities of the solution, Lett. Math. Phys. 105 (2015), no. 6, 795–825.

    Article  MathSciNet  MATH  Google Scholar 

  64. M. Shifman, Resurgence, operator product expansion, and remarks on renormalons in supersymmetric Yang-Mills theory, J. Exp. Theor. Phys. 120 (2015), no. 3, 386–398. [arXiv:1411.4004]

    Article  Google Scholar 

  65. G. V. Dunne, M. Shifman and M. Ünsal, Infrared renormalons versus operator product expansions in supersymmetric and related Gauge theories, Phys. Rev. Lett. 114 (2015), no. 19, 191601. [arXiv:1502.06680]

    Article  Google Scholar 

  66. A. Behtash, E. Poppitz, T. Sulejmanpasic and M. Ünsal, The curious incident of multi-instantons and the necessity of Lefschetz thimbles, JHEP 11 (2015), 175. [arXiv:1507.04063]

    Article  MathSciNet  Google Scholar 

  67. M. Mariño, Open string amplitudes and large-order behavior in topological string theory, JHEP 0803 (2008), 060. [arXiv:hep-th/0612127]

    Article  MathSciNet  Google Scholar 

  68. B. Eynard and M. Mariño, A Holomorphic and background independent partition function for matrix models and topological strings, J. Geom. Phys. 61 (2011), 1181–1202. [arXiv:0810.4273]

    Article  MathSciNet  MATH  Google Scholar 

  69. A. Klemm, M. Mariño and M. Rauch, Direct integration and non-perturbative effects in matrix models, JHEP 1010 (2010), 004. [arXiv:1002.3846]

    Article  MathSciNet  MATH  Google Scholar 

  70. N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 1111 (2011), 141. [arXiv:1103.4844]

    Article  MathSciNet  MATH  Google Scholar 

  71. R. Couso-Santamaría, J. D. Edelstein, R. Schiappa and M. Vonk, Resurgent transseries and the holomorphic anomaly, Annales Henri Poincaré, in press (2013). [arXiv:1308.1695]

    Google Scholar 

  72. A. Grassi, M. Mariño and S. Zakany, Resumming the string perturbation series, JHEP 1505 (2015), 038. [arXiv:1405.4214]

    Article  MathSciNet  Google Scholar 

  73. R. Couso-Santamaría, J. D. Edelstein, R. Schiappa and M. Vonk, Resurgent transseries and the holomorphic anomaly: nonperturbative closed strings in local ℂℙ2, Commun. Math. Phys. 338 (2015), no. 1, 285–346. [arXiv:1407.4821]

    Article  MathSciNet  MATH  Google Scholar 

  74. I. Muller, Zum Paradoxon der Warmeleitungstheorie, Z. Phys. 198 (1967), 329–344.

    Article  MATH  Google Scholar 

  75. W. Israel and J. Stewart, Transient relativistic thermodynamics and kinetic theory, Annals Phys. 118 (1979), 341–372.

    Article  MathSciNet  Google Scholar 

  76. E. Delabaere, Introduction to the Écalle theory, In: “Computer Algebra and Differential Equations”, E. Tournier, (ed.), Cambridge University Press, 1994, 59–102.

    Google Scholar 

  77. O. Costin, “Asymptotics and Borel Summability”, Monographs and Surveys in Pure and Applied Mathematics, Chapman and Hall/CRC, 2008.

    Google Scholar 

  78. J. Zinn-Justin, Instantons in quantum mechanics: numerical evidence for a conjecture, J. Math. Phys. 25 (1984), 549.

    Article  MathSciNet  Google Scholar 

  79. M. V. Berry and C. J. Howls, Hyperasymptotics, Proc. R. Soc. London A430 (1990), 653–668.

    Article  MathSciNet  MATH  Google Scholar 

  80. M. V. Berry and C. J. Howls, Hyperasymptotics for integrals with saddles, Proc. R. Soc. London A434 (1991), 657.

    Article  MathSciNet  MATH  Google Scholar 

  81. M. V. Berry, “Asymptotics, Superasymptotics, Hyperasymptotics…” Asymptotics beyond all orders, Plenum, New York, 1991.

    Google Scholar 

  82. J. C. Collins and D. E. Soper, Large order expansion in perturbation theory, Annals Phys. 112 (1978), 209–234.

    Article  MathSciNet  Google Scholar 

  83. T. Misumi, M. Nitta and N. Sakai, Non-BPS exact solutions and their relation to bions in ℂPN−1 models, JHEP 05 (2016), 057. [arXiv:1604.00839]

    Article  Google Scholar 

  84. M. P. Heller, R. A. Janik and P. Witaszczyk, Hydrodynamic gradient expansion in Gauge theory plasmas, Phys. Rev. Lett. 110 (2013), no. 21, 211602. [arXiv:1302.0697]

    Article  Google Scholar 

  85. S. Demulder, D. Dorigoni and D. C. Thompson, Resurgence in η-deformed principal chiral models, JHEP 07 (2016), 088. [arXiv:1604.07851]

    Article  Google Scholar 

  86. F. Pham, Vanishing homologies and the n variable saddle-point method, Proc. Sympos. Pure Math. 40 (1983), 319.

    Article  Google Scholar 

  87. E. Delabaere and C. J. Howls, Global asymptotics for multiple integrals with boundaries, Duke Math. J. 112 (2002), 199–264.

    Article  MathSciNet  MATH  Google Scholar 

  88. C. J. Howls, P. J. Langman and A. B. O. Daalhuis, On the higher-order Stokes phenomenon, Proc. R. Soc. London A460 (2004), 2285.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Authors
  1. Inês Aniceto
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Frédéric Fauvet Dominique Manchon Stefano Marmi David Sauzin

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Scuola Normale Superiore Pisa

About this paper

Cite this paper

Aniceto, I. (2017). Asymptotics, ambiguities and resurgence. In: Fauvet, F., Manchon, D., Marmi, S., Sauzin, D. (eds) Resurgence, Physics and Numbers. CRM Series, vol 20. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-613-1_1

Download citation

Publish with us

Policies and ethics

Search

Navigation