Journal of High Energy Physics

, 2015:160 | Cite as

Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems

Open Access
Regular Article - Theoretical Physics


The Nekrasov-Shatashvili limit for the low-energy behavior of \( \mathcal{N}=2 \) and \( \mathcal{N}={2}^{*} \) supersymmetric SU(2) gauge theories is encoded in the spectrum of the Mathieu and Lamé equations, respectively. This correspondence is usually expressed via an all-orders Bohr-Sommerfeld relation, but this neglects non-perturbative effects, the nature of which is very different in the electric, magnetic and dyonic regions. In the gauge theory dyonic region the spectral expansions are divergent, and indeed are not Borel-summable, so they are more properly described by resurgent trans-series in which perturbative and non-perturbative effects are deeply entwined. In the gauge theory electric region the spectral expansions are convergent, but nevertheless there are non-perturbative effects due to poles in the expansion coefficients, and which we associate with worldline instantons. This provides a concrete analog of a phenomenon found recently by Drukker, Mariño and Putrov in the large N expansion of the ABJM matrix model, in which non-perturbative effects are related to complex space-time instantons. In this paper we study how these very different regimes arise from an exact WKB analysis, and join smoothly through the magnetic region. This approach also leads to a simple proof of a resurgence relation found recently by Dunne and Ünsal, showing that for these spectral systems all non-perturbative effects are subtly encoded in perturbation theory, and identifies this with the Picard-Fuchs equation for the quantized elliptic curve.


Nonperturbative Effects Supersymmetric gauge theory 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    J. Écalle, Les fonctions resurgentes, volumes I–III, Publ. Math. Orsay, France (1981).Google Scholar
  2. [2]
    O. Costin, Asymptotics and Borel summability, Chapman & Hall/CRC, U.S.A. (2009).MATHGoogle Scholar
  3. [3]
    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] [INSPIRE].CrossRefMATHGoogle Scholar
  4. [4]
    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] [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  5. [5]
    I. Aniceto, R. Schiappa and M. Vonk, The resurgence of instantons in string theory, Commun. Num. Theor. Phys. 6 (2012) 339 [arXiv:1106.5922] [INSPIRE].CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    M. Mariño, Lectures on non-perturbative effects in large-N gauge theories, matrix models and strings, Fortsch. Phys. 62 (2014) 455 [arXiv:1206.6272] [INSPIRE].CrossRefADSGoogle Scholar
  7. [7]
    P. Argyres and M. Ünsal, A semiclassical realization of infrared renormalons, Phys. Rev. Lett. 109 (2012) 121601 [arXiv:1204.1661] [INSPIRE].CrossRefADSGoogle Scholar
  8. [8]
    P.C. Argyres and M. Ünsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion and renormalon effects, JHEP 08 (2012) 063 [arXiv:1206.1890] [INSPIRE].CrossRefADSGoogle Scholar
  9. [9]
    G.V. Dunne and M. Ünsal, Resurgence and trans-series in quantum field theory: the CP(N − 1) model, JHEP 11 (2012) 170 [arXiv:1210.2423] [INSPIRE].CrossRefADSGoogle Scholar
  10. [10]
    G.V. Dunne and M. Ünsal, Continuity and resurgence: towards a continuum definition of the ℂℙ(N − 1) model, Phys. Rev. D 87 (2013) 025015 [arXiv:1210.3646] [INSPIRE].ADSGoogle Scholar
  11. [11]
    G.V. Dunne and M. Ünsal, Generating nonperturbative physics from perturbation theory, Phys. Rev. D 89 (2014) 041701 [arXiv:1306.4405] [INSPIRE].ADSGoogle Scholar
  12. [12]
    G.V. Dunne and M. Ünsal, Uniform WKB, multi-instantons and resurgent trans-series, Phys. Rev. D 89 (2014) 105009 [arXiv:1401.5202] [INSPIRE].ADSGoogle Scholar
  13. [13]
    I. Aniceto and R. Schiappa, Nonperturbative ambiguities and the reality of resurgent transseries, arXiv:1308.1115 [INSPIRE].
  14. [14]
    I. Aniceto, J.G. Russo and R. Schiappa, Resurgent analysis of localizable observables in supersymmetric gauge theories, arXiv:1410.5834 [INSPIRE].
  15. [15]
    D. Dorigoni, An introduction to resurgence, trans-series and alien calculus, arXiv:1411.3585 [INSPIRE].
  16. [16]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
  17. [17]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  19. [19]
    A. Bilal, Duality in N = 2 SUSY SU(2) Yang-Mills theory: A Pedagogical introduction to the work of Seiberg and Witten, in the proceedings of Quantum fields and quantum space time, G. ’t Hooft et al. eds., Plenum Press, U.S.A. (1997), hep-th/9601007 [INSPIRE].
  20. [20]
    W. Lerche, Introduction to Seiberg-Witten theory and its stringy origin, Nucl. Phys. Proc. Suppl. 55B (1997) 83 [Fortsch. Phys. 45 (1997) 293] [hep-th/9611190] [INSPIRE].
  21. [21]
    L. Álvarez-Gaumé and S.F. Hassan, Introduction to S duality in N = 2 supersymmetric gauge theories: a pedagogical review of the work of Seiberg and Witten, Fortsch. Phys. 45 (1997) 159 [hep-th/9701069] [INSPIRE].CrossRefADSMATHGoogle Scholar
  22. [22]
    N. Dorey, T.J. Hollowood, V.V. Khoze and M.P. Mattis, The calculus of many instantons, Phys. Rept. 371 (2002) 231 [hep-th/0206063] [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  23. [23]
    J. Teschner, Exact results on N = 2 supersymmetric gauge theories, arXiv:1412.7145 [INSPIRE].
  24. [24]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  25. [25]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in the proceedings of 16th International Congress on Mathematical Physics (ICMP09), P. Exner ed., World Scientific, Singapore (2010), arXiv:0908.4052 [INSPIRE].
  26. [26]
    N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].CrossRefADSMATHGoogle Scholar
  27. [27]
    N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  28. [28]
    E. Langmann, Explicit solution of the (quantum) elliptic Calogero-Sutherland model, Ann. Henri Poincare 15 (2014) 755 [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  29. [29]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  30. [30]
    V.A. Fateev and A.V. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  31. [31]
    A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  32. [32]
    A. Mironov and A. Morozov, Nekrasov functions from exact BS periods: the case of SU(N), J. Phys. A 43 (2010) 195401 [arXiv:0911.2396] [INSPIRE].ADSMathSciNetGoogle Scholar
  33. [33]
    K. Maruyoshi and M. Taki, Deformed prepotential, quantum integrable system and Liouville field theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  34. [34]
    W. He and Y.-G. Miao, Magnetic expansion of Nekrasov theory: the SU(2) pure gauge theory, Phys. Rev. D 82 (2010) 025020 [arXiv:1006.1214] [INSPIRE].ADSGoogle Scholar
  35. [35]
    W. He, Combinatorial approach to Mathieu and Lamé equations, arXiv:1108.0300 [INSPIRE].
  36. [36]
    W. He, Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations, Annals Phys. 353 (2015) 150 [arXiv:1401.4135] [INSPIRE].CrossRefADSGoogle Scholar
  37. [37]
    M.X. Huang, A.K. Kashani-Poor and A. Klemm, The Ω deformed B-model for rigid N = 2 theories, Ann. Henri Poincare 14 (2013) 425 [arXiv:1109.5728] [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  38. [38]
    M.X. Huang, On gauge theory and topological string in Nekrasov-Shatashvili limit, JHEP 06 (2012) 152 [arXiv:1205.3652] [INSPIRE].CrossRefADSGoogle Scholar
  39. [39]
    A.-K. Kashani-Poor and J. Troost, The toroidal block and the genus expansion, JHEP 03 (2013) 133 [arXiv:1212.0722] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  40. [40]
    A.-K. Kashani-Poor and J. Troost, Quantum geometry from the toroidal block, JHEP 08 (2014) 117 [arXiv:1404.7378] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  41. [41]
    M. Piatek, Classical torus conformal block, N = 2* twisted superpotential and the accessory parameter of Lamé equation, JHEP 03 (2014) 124 [arXiv:1309.7672] [INSPIRE].CrossRefADSGoogle Scholar
  42. [42]
    M. Piatek and A.R. Pietrykowski, Classical irregular block, \( \mathcal{N}=2 \) pure gauge theory and Mathieu equation, JHEP 12 (2014) 032 [arXiv:1407.0305] [INSPIRE].CrossRefADSGoogle Scholar
  43. [43]
    D. Krefl, Non-perturbative quantum geometry, JHEP 02 (2014) 084 [arXiv:1311.0584] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  44. [44]
    D. Krefl, Non-perturbative quantum geometry II, JHEP 12 (2014) 118 [arXiv:1410.7116] [INSPIRE].CrossRefADSGoogle Scholar
  45. [45]
    A. Gorsky and A. Milekhin, RG-Whitham dynamics and complex Hamiltonian systems, arXiv:1408.0425 [INSPIRE].
  46. [46]
    J. Zinn-Justin and U.D. Jentschura, Multi-instantons and exact results I: Conjectures, WKB expansions and instanton interactions, Annals Phys. 313 (2004) 197 [quant-ph/0501136] [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  47. [47]
    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 [quant-ph/0501137] [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  48. [48]
    G. Basar, G.V. Dunne and M. Ünsal, Resurgence theory, ghost-instantons and analytic continuation of path integrals, JHEP 10 (2013) 041 [arXiv:1308.1108] [INSPIRE].CrossRefADSGoogle Scholar
  49. [49]
    N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].CrossRefADSMATHGoogle Scholar
  50. [50]
    N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [INSPIRE].CrossRefADSGoogle Scholar
  51. [51]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Exact results on the ABJM Fermi gas, JHEP 10 (2012) 020 [arXiv:1207.4283] [INSPIRE].CrossRefADSGoogle Scholar
  52. [52]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Instanton effects in ABJM theory from Fermi gas approach, JHEP 01 (2013) 158 [arXiv:1211.1251] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  53. [53]
    J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, arXiv:1308.6485 [INSPIRE].
  54. [54]
    J. Kallen, The spectral problem of the ABJ Fermi gas, arXiv:1407.0625 [INSPIRE].
  55. [55]
    G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin systems via beta-deformed matrix models, arXiv:1104.4016 [INSPIRE].
  56. [56]
    M. Aganagic, M.C.N. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum geometry of refined topological strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  57. [57]
    T. Gulden, M. Janas, P. Koroteev and A. Kamenev, Statistical mechanics of Coulomb gases as quantum theory on Riemann surfaces, Zh. Eksp. Teor. Fiz. 144 (2013) 574 [J. Exp. Theor. Phys. 117 (2013) 517] [arXiv:1303.6386] [INSPIRE].
  58. [58]
    A. Cherman, P. Koroteev and M. Ünsal, Resurgence and holomorphy: from weak to strong coupling, arXiv:1410.0388 [INSPIRE].
  59. [59]
    J.L. Dunham, The Wentzel-Brillouin-Kramers method of solving the wave equation, Phys. Rev. 41 (1932) 713.CrossRefADSGoogle Scholar
  60. [60]
    C.M. Bender and S.A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill, New York U.S.A. (1978).MATHGoogle Scholar
  61. [61]
    A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. Inst. H. Poincaré 39 (1983) 211.MATHMathSciNetGoogle Scholar
  62. [62]
    A. Voros, Zeta-regularisation for exact-WKB resolution of a general 1D Schrödinger equation, arXiv:1202.3100 [INSPIRE].
  63. [63]
    T. Kawai and Y. Takei, Secular equations through the exact WKB analysis, RIMS, Kyoto University, Japan (1991).Google Scholar
  64. [64]
    T. Aoki, T. Kawai and Y. Takei, Algebraic analysis of singular perturbations: On exact WKB analysis, RIMS-947 (1993) [INSPIRE].
  65. [65]
    E. Delabaere, Spectre de lopérateur de Schrödinger stationnaire unidimensionnel à potentiel polynôme trigonométrique, C. R. Acad. Sci. Paris 314 (1992) 807.MATHMathSciNetGoogle Scholar
  66. [66]
    E. Delabaere, H. Dillinger and F. Pham, Exact semiclassical expansions for one-dimensional quantum oscillators, J. Math. Phys. 38 (1997) 6126.CrossRefADSMATHMathSciNetGoogle Scholar
  67. [67]
    E. Delabaere and F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré 71 (1999) 1.MATHMathSciNetGoogle Scholar
  68. [68]
    C.J. Howls, T. Kawai and Y. Takei, Toward the exact WKB analysis of differential equations, linear or non-linear, Kyoto University Press, Japan (2000).MATHGoogle Scholar
  69. [69]
    L.V. Keldysh, Ionization in the field of a strong electromagnetic wave, Sov. Phys. JETP 20 (1965) 1307.MathSciNetGoogle Scholar
  70. [70]
    W. Heisenberg and H. Euler, Consequences of Diracs theory of positrons, Z. Phys. 98 (1936) 714 [physics/0605038] [INSPIRE].CrossRefADSGoogle Scholar
  71. [71]
    J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  72. [72]
    G.V. Dunne, Heisenberg-Euler effective Lagrangians: basics and extensions, in Ian Kogan Memorial Collection. From fields to strings: circumnavigating theoretical physics, M. Shifman et al. ed., World Scientific, Singapore (2005), hep-th/0406216 [INSPIRE].
  73. [73]
    E. Brézin and C. Itzykson, Pair production in vacuum by an alternating field, Phys. Rev. D 2 (1970) 1191 [INSPIRE].ADSGoogle Scholar
  74. [74]
    V.S. Popov, Pair production in a variable external field (quasiclassical approximation), Sov. Phys. JETP 34 (1972) 709.ADSGoogle Scholar
  75. [75]
    M.S. Marinov and V.S. Popov, Electron-positron pair creation from vacuum induced by variable electric field, Fortsch. Phys. 25 (1977) 373 [INSPIRE].CrossRefADSGoogle Scholar
  76. [76]
    NIST digital library of mathematical functions,
  77. [77]
    E.T. Whittaker and G.N. Watson, A course of modern analysis, Cambridge University Press, Cambridge U.K. (1902).MATHGoogle Scholar
  78. [78]
    J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen, Springer-Verlag, Berlin Germany (1954).CrossRefGoogle Scholar
  79. [79]
    W. Magnus and S. Winkler, Hills equation, John Wiley & Sons, New York U.S.A. (1966).MATHGoogle Scholar
  80. [80]
    R.B. Dingle and H.J.W. Müller, Asymptotic expansions of Mathieu functions and their characteristic numbers, J. Reine Angew. Math. 211 (1962) 11.MATHMathSciNetGoogle Scholar
  81. [81]
    R.E. Peierls, Quantum theory of solids, Clarendon Press, Oxford U.K. (1996).Google Scholar
  82. [82]
    E.B. Bogomolny, Calculation of instanton-anti-instanton contributions in quantum mechanics, Phys. Lett. B 91 (1980) 431 [INSPIRE].CrossRefADSGoogle Scholar
  83. [83]
    J. Zinn-Justin, Multi-instanton contributions in quantum mechanics, Nucl. Phys. B 192 (1981) 125 [INSPIRE].CrossRefADSGoogle Scholar
  84. [84]
    I.I. Balitsky and A.V. Yung, Instanton molecular vacuum in N = 1 supersymmetric quantum mechanics, Nucl. Phys. B 274 (1986) 475 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  85. [85]
    J.C. Le Guillou and J. Zinn-Justin, Large order behavior of perturbation theory, North-Holland, Amsterdam The Netherlands (1990).Google Scholar
  86. [86]
    J. Zinn-Justin, Quantum field theory and critical phenomena, Oxford University Press, Oxford U.K. (2002).CrossRefGoogle Scholar
  87. [87]
    K. Konishi and G. Paffuti, Quantum mechanics: a new introduction, Oxford University Press, Oxford U.K. (2009).Google Scholar
  88. [88]
    L.D. Landau and E.M. Lifshitz, Quantum mechanics (non-relativistic theory), Elsevier (2003).Google Scholar
  89. [89]
    A.M. Dykhne, Quasiclassical particles in a one-dimensional periodic potential, Sov. Phys. JETP 13 (1961) 999 [J. Exp. Theor. Phys. 40 (1961) 1423].Google Scholar
  90. [90]
    H. Neuberger, Semiclassical calculation of the energy dispersion relation in the valence band of the quantum pendulum, Phys. Rev. D 17 (1978) 498 [INSPIRE].ADSMathSciNetGoogle Scholar
  91. [91]
    N. Fröman, Dispersion relation for energy bands and energy gaps derived by the use of a phase-integral method, with an application to the Mathieu equation, J. Phys. A 12 (1979) 2355.ADSGoogle Scholar
  92. [92]
    J.N.L. Connor, T. Uzer, R.A. Marcus and A.D. Smith, Eigenvalues of the Schrödinger equation for a periodic potential with nonperiodic boundary conditions: A uniform semiclassical analysis, J. Chem. Phys. 80 (1984) 5095CrossRefADSMathSciNetGoogle Scholar
  93. [93]
    M.I. Weinstein and J.B. Keller, Hills equation with a large potential, SIAM J. Appl. Math. 45 (1985) 200.CrossRefMATHMathSciNetGoogle Scholar
  94. [94]
    M.I. Weinstein and J.B. Keller, Asymptotic behavior of stability regions for Hills equation, SIAM J. Appl. Math. 47 (1987) 941.CrossRefMATHMathSciNetGoogle Scholar
  95. [95]
    A. Klemm, W. Lerche and S. Theisen, Nonperturbative effective actions of N = 2 supersymmetric gauge theories, Int. J. Mod. Phys. A 11 (1996) 1929 [hep-th/9505150] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  96. [96]
    M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett. B 357 (1995) 342 [hep-th/9506102] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  97. [97]
    A. Bilal and F. Ferrari, Curves of marginal stability and weak and strong coupling BPS spectra in N = 2 supersymmetric QCD, Nucl. Phys. B 480 (1996) 589 [hep-th/9605101] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  98. [98]
    R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  99. [99]
    H.E. Fettis, On the reciprocal modulus relation for elliptic integrals, SIAM J. Math. Anal. 1 (1970) 524.CrossRefMATHMathSciNetGoogle Scholar
  100. [100]
    D. Gaiotto, Asymptotically free \( \mathcal{N}=2 \) theories and irregular conformal blocks, J. Phys. Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].CrossRefGoogle Scholar
  101. [101]
    R. Balian, G. Parisi and A. Voros, Discrepancies from asymptotic series and their relation to complex classical trajectories, Phys. Rev. Lett. 41 (1978) 1141.CrossRefADSGoogle Scholar
  102. [102]
    R. Balian, G. Parisi and A. Voros, Quartic Oscillator, in Marseille 1978, Proceedings, Feynman Path Integrals, Berlin, Germany (1979).Google Scholar
  103. [103]
    G.V. Dunne and C. Schubert, Worldline instantons and pair production in inhomogeneous fields, Phys. Rev. D 72 (2005) 105004 [hep-th/0507174] [INSPIRE].ADSMathSciNetGoogle Scholar
  104. [104]
    G.V. Dunne, Q.-h. Wang, H. Gies and C. Schubert, Worldline instantons. II. The fluctuation prefactor, Phys. Rev. D 73 (2006) 065028 [hep-th/0602176] [INSPIRE].
  105. [105]
    C.K. Dumlu and G.V. Dunne, The Stokes phenomenon and Schwinger vacuum pair production in time-dependent laser pulses, Phys. Rev. Lett. 104 (2010) 250402 [arXiv:1004.2509] [INSPIRE].CrossRefADSGoogle Scholar
  106. [106]
    C.K. Dumlu and G.V. Dunne, Complex worldline instantons and quantum interference in vacuum pair production, Phys. Rev. D 84 (2011) 125023 [arXiv:1110.1657] [INSPIRE].ADSGoogle Scholar
  107. [107]
    G. Basar, G. V. Dunne and M. Ünsal, in preparation.Google Scholar
  108. [108]
    H. Volkmer, Four remarks on eigenvalues of Lamés equation, Anal. Appl. 2 (2004) 161.CrossRefMATHMathSciNetGoogle Scholar
  109. [109]
    H.J.W. Müller, On asymptotic expansions of ellipsoidal wave functions, Math. Nachrichten 32 (1966) 157.CrossRefMATHGoogle Scholar
  110. [110]
    G.V. Dunne and K. Rao, Lamé instantons, JHEP 01 (2000) 019 [hep-th/9906113] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  111. [111]
    G.V. Dunne and M. Shifman, Duality and selfduality (energy reflection symmetry) of quasiexactly solvable periodic potentials, Annals Phys. 299 (2002) 143 [hep-th/0204224] [INSPIRE].CrossRefADSMATHMathSciNetGoogle Scholar
  112. [112]
    I.M. Gelfand and L.A. Dikii, Asymptotic behavior of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-De Vries equations, Russ. Math. Surveys 30 (1975) 77 [Usp. Mat. Nauk 30 (1975) 67] [INSPIRE].
  113. [113]
    A.M. Perelomov and Y.B. Zeldovich, Quantum mechanics: selected topics, World Scientific, Singapore (1998).CrossRefMATHGoogle Scholar
  114. [114]
    M.P. Grosset, A.P. Veselov, Elliptic Faulhaber polynomials and Lamé densities of states, math-ph/0508066.
  115. [115]
    D.B. Fairlie and A.P. Veselov, Faulhaber and Bernoulii polynomials and solitons, Physica D 152 (2001) 47.ADSMathSciNetGoogle Scholar
  116. [116]
    N. Dorey, V.V. Khoze and M.P. Mattis, On mass deformed N = 4 supersymmetric Yang-Mills theory, Phys. Lett. B 396 (1997) 141 [hep-th/9612231] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  117. [117]
    J.A. Minahan, D. Nemeschansky and N.P. Warner, Instanton expansions for mass deformed N = 4 super Yang-Mills theories, Nucl. Phys. B 528 (1998) 109 [hep-th/9710146] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  118. [118]
    M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, Deformed N = 2 theories, generalized recursion relations and S-duality, JHEP 04 (2013) 039 [arXiv:1302.0686] [INSPIRE].CrossRefADSGoogle Scholar
  119. [119]
    M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, Modular anomaly equation, heat kernel and S-duality in N = 2 theories, JHEP 11 (2013) 123 [arXiv:1307.6648] [INSPIRE].CrossRefADSGoogle Scholar
  120. [120]
    M. Billó et al., Modular anomaly equations in \( \mathcal{N}={2}^{\ast } \) theories and their large-N limit, JHEP 10 (2014) 131 [arXiv:1406.7255] [INSPIRE].CrossRefADSGoogle Scholar
  121. [121]
    R.P. Stanley, Enumerative combinatorics, volume 2, Cambridge University Press, Cambridge U.K. (2001).Google Scholar
  122. [122]
    B.A. Dubrovin, Inverse problem for periodic finite-zoned potentials in the theory of scattering, Funktsionaln. Analiz i ego Prilozhenija 9 (1975) 65 [Funct. Anal. Appl. 9 (1975) 61].Google Scholar
  123. [123]
    M. Stone and J. Reeve, Late terms in the asymptotic expansion for the energy levels of a periodic potential, Phys. Rev. D 18 (1978) 4746 [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Maryland Center for Fundamental PhysicsUniversity of MarylandCollege ParkU.S.A.
  2. 2.Department of PhysicsUniversity of ConnecticutStorrsU.S.A.

Personalised recommendations