Exact WKB analysis of \( \mathcal{N} \) = 2 gauge theories

  • Sujay K. Ashok
  • Dileep P. Jatkar
  • Renjan R. John
  • M. Raman
  • Jan Troost
Open Access
Regular Article - Theoretical Physics

Abstract

We study \( \mathcal{N} \) = 2 supersymmetric gauge theories with gauge group SU(2) coupled to fundamental flavours, covering all asymptotically free and conformal cases. We re-derive, from the conformal field theory perspective, the differential equations satisfied by ϵ1- and ϵ2-deformed instanton partition functions. We confirm their validity at leading order in ϵ2 via a saddle-point analysis of the partition function. In the semi-classical limit we show that these differential equations take a form amenable to exact WKB analysis. We compute the monodromy group associated to the differential equations in terms of ϵ1-deformed and Borel resummed Seiberg-Witten data. For each case, we study pairs of Stokes graphs that are related by flips and pops, and show that the monodromy groups allow one to confirm the Stokes automorphisms that arise as the phase of ϵ1 is varied. Finally, we relate the Borel resummed monodromies with the traditional Seiberg-Witten variables in the semi-classical limit.

Keywords

Extended Supersymmetry Solitons Monopoles and Instantons Supersymmetric gauge theory 

Notes

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.

References

  1. [1]
    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].
  2. [2]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    A. Marshakov, A. Mironov and A. Morozov, On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    J. Teschner and G.S. Vartanov, Supersymmetric gauge theories, quantization of \( {\mathrm{\mathcal{M}}}_{\mathrm{flat}} \) and conformal field theory, Adv. Theor. Math. Phys. 19 (2015) 1 [arXiv:1302.3778] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String, Adv. Theor. Math. Phys. 16 (2012) 725 [arXiv:1008.0574] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
  17. [17]
    A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-Zommerfeld Integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    V.A. Fateev and A.V. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    A.-K. Kashani-Poor and J. Troost, The toroidal block and the genus expansion, JHEP 03 (2013) 133 [arXiv:1212.0722] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama, Non-perturbative effects and the refined topological string, JHEP 09 (2014) 168 [arXiv:1306.1734] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, Annales Henri Poincaré 17 (2016) 1037 [arXiv:1308.6485] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    A. Voros, The return of the quartic oscillator: the complex WKB method, Ann. Inst. H. Poincaré A 39 (1983) 211.MathSciNetMATHGoogle Scholar
  23. [23]
    H. Dillinger, E. Delabaere and F. Pham, Résurgence de voros et périodes des courbes hyperelliptiques, Annales de l’institut Fourier 43 (1993) 163.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    A.O. Jidoumou, Modèles de résurgence paramétrique: fonctions d’airy et cyclindro-paraboliques, Journal de mathématiques pures et appliquées 73 (1994) 111.MathSciNetMATHGoogle Scholar
  25. [25]
    E. Delabaere, H. Dillinger and F. Pham, Exact semiclassical expansions for one-dimensional quantum oscillators, J. Math. Phys. 38 (1997) 6126.ADSMathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    E. Delabaere and F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré A 71 (1999) 1.MathSciNetMATHGoogle Scholar
  27. [27]
    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].
  28. [28]
    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].
  29. [29]
    T. Kawai and Y. Takei, Algebraic analysis of singular perturbation theory, volume 227, American Mathematical Soc., U.S.A. (2005).Google Scholar
  30. [30]
    O. Costin, Asymptotics and Borel summability, CRC Press, (2008).Google Scholar
  31. [31]
    K. Iwaki and T. Nakanishi, Exact wkb analysis and cluster algebras, J. Phys. A 47 (2014) 474009.MathSciNetMATHGoogle Scholar
  32. [32]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  33. [33]
    A.-K. Kashani-Poor and J. Troost, Pure \( \mathcal{N} \) = 2 super Yang-Mills and exact WKB, JHEP 08 (2015) 160 [arXiv:1504.08324] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  34. [34]
    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
  35. [35]
    N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
  36. [36]
    A.-K. Kashani-Poor and J. Troost, Transformations of Spherical Blocks, JHEP 10 (2013) 009 [arXiv:1305.7408] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    S.K. Ashok, M. Billó, E. Dell’Aquila, M. Frau, R.R. John and A. Lerda, Non-perturbative studies of N = 2 conformal quiver gauge theories, Fortsch. Phys. 63 (2015) 259 [arXiv:1502.05581] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    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
  39. [39]
    W. He and Y.-G. Miao, Mathieu equation and Elliptic curve, Commun. Theor. Phys. 58 (2012) 827 [arXiv:1006.5185] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  40. [40]
    G. Basar and G.V. Dunne, Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems, JHEP 02 (2015) 160 [arXiv:1501.05671] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435 [INSPIRE].
  42. [42]
    D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi and C. Vafa, \( \mathcal{N} \) = 2 quantum field theories and their BPS quivers, Adv. Theor. Math. Phys. 18 (2014) 27 [arXiv:1112.3984] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    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].ADSCrossRefGoogle Scholar
  45. [45]
    W. He, Quasimodular instanton partition function and the elliptic solution of Korteweg-de Vries equations, Annals Phys. 353 (2015) 150 [arXiv:1401.4135] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    A.-K. Kashani-Poor and J. Troost, Quantum geometry from the toroidal block, JHEP 08 (2014) 117 [arXiv:1404.7378] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  47. [47]
    M. Billó, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential in \( \mathcal{N} \) = 2 theories (I): the ADE algebras, JHEP 11 (2015) 024 [arXiv:1507.07709] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  48. [48]
    M. Billó, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential of \( \mathcal{N} \) = 2 theories (II): the non-simply laced algebras, JHEP 11 (2015) 026 [arXiv:1507.08027] [INSPIRE].
  49. [49]
    S.K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, A. Lerda and M. Raman, Modular anomaly equations and S-duality in \( \mathcal{N} \) = 2 conformal SQCD, JHEP 10 (2015) 091 [arXiv:1507.07476] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    S.K. Ashok, E. Dell’Aquila, A. Lerda and M. Raman, S-duality, triangle groups and modular anomalies in \( \mathcal{N} \) = 2 SQCD, JHEP 04 (2016) 118 [arXiv:1601.01827] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    N. Wyllard, A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N ) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    R. Poghossian, Deformed SW curve and the null vector decoupling equation in Toda field theory, JHEP 04 (2016) 070 [arXiv:1601.05096] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    D. Gaiotto, G.W. Moore and A. Neitzke, Spectral networks, Annales Henri Poincaré 14 (2013) 1643 [arXiv:1204.4824] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  54. [54]
    D. Gaiotto, G.W. Moore and A. Neitzke, Spectral Networks and Snakes, Annales Henri Poincaré 15 (2014) 61 [arXiv:1209.0866] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  55. [55]
    R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    F. Fucito, J.F. Morales, D.R. Pacifici and R. Poghossian, Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves, JHEP 05 (2011) 098 [arXiv:1103.4495] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  57. [57]
    F. Fucito, J.F. Morales and D. Ricci Pacifici, Deformed Seiberg-Witten Curves for ADE Quivers, JHEP 01 (2013) 091 [arXiv:1210.3580] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Sujay K. Ashok
    • 1
  • Dileep P. Jatkar
    • 2
  • Renjan R. John
    • 1
  • M. Raman
    • 1
  • Jan Troost
    • 3
  1. 1.Institute of Mathematical SciencesChennaiIndia
  2. 2.Harish-Chandra Research InstituteJhusiIndia
  3. 3.Laboratoire de Physique Théorique de l’ École Normale Supérieure, CNRS, PSL Research University, Sorbonne UniversitésParisFrance

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