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Exact WKB analysis of ℂℙ1 holomorphic blocks

  • Sujay K. AshokEmail author
  • P. N. Bala Subramanian
  • Aditya Bawane
  • Dharmesh Jain
  • Dileep P. Jatkar
  • Arkajyoti Manna
Open Access
Regular Article - Theoretical Physics
  • 27 Downloads

Abstract

We study holomorphic blocks in the three dimensional \( \mathcal{N} \) = 2 gauge theory that describes the ℂℙ1 model. We apply exact WKB methods to analyze the line operator identities associated to the holomorphic blocks and derive the analytic continuation formulae of the blocks as the twisted mass and FI parameter are varied. The main technical result we utilize is the connection formula for the 1𝜙1q-hypergeometric function. We show in detail how the q-Borel resummation methods reproduce the results obtained previously by using block-integral methods.

Keywords

Supersymmetric Gauge Theory Chern-Simons Theories Duality in Gauge Field Theories Nonperturbative Effects 

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]
    K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett.B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
  2. [2]
    O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys.B 499 (1997) 67 [hep-th/9703110] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    N. Dorey and D. Tong, Mirror symmetry and toric geometry in three-dimensional gauge theories, JHEP05 (2000) 018 [hep-th/9911094] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    D. Tong, Dynamics of N = 2 supersymmetric Chern-Simons theories, JHEP07 (2000) 019 [hep-th/0005186] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    E. Witten, Topological quantum field theory, Commun. Math. Phys.117 (1988) 353 [INSPIRE].ADSMathSciNetCrossRefGoogle 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].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP03 (2010) 089 [arXiv:0909.4559] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP03 (2011) 127 [arXiv:1012.3512] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
  11. [11]
    Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev.D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
  12. [12]
    J. Nian, Localization of supersymmetric Chern-Simons-matter theory on a squashed S 3with SU(2) × U(1) isometry, JHEP07 (2014) 126 [arXiv:1309.3266] [INSPIRE].
  13. [13]
    V. Pestun and M. Zabzine, Introduction to localization in quantum field theory, J. Phys.A 50 (2017) 443001 [arXiv:1608.02953] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  14. [14]
    S. Pasquetti, Factorisation of N = 2 theories on the squashed 3-sphere, JHEP04 (2012) 120 [arXiv:1111.6905] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys.17 (2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  16. [16]
    C. Beem, T. Dimofte and S. Pasquetti, Holomorphic blocks in three dimensions, JHEP12 (2014) 177 [arXiv:1211.1986] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    A. Nedelin, S. Pasquetti and Y. Zenkevich, T [SU(N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences, JHEP02 (2019) 176 [arXiv:1712.08140] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    F. Aprile, S. Pasquetti and Y. Zenkevich, Flipping the head of T [SU(N)]: mirror symmetry, spectral duality and monopoles, JHEP04 (2019) 138 [arXiv:1812.08142] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    Y. Imamura, H. Matsuno and D. Yokoyama, Factorization of the S 3/Z npartition function, Phys. Rev.D 89 (2014) 085003 [arXiv:1311.2371] [INSPIRE].
  20. [20]
    F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4d, JHEP11 (2015) 155 [arXiv:1507.00261] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    C. Closset, H. Kim and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP03 (2017) 074 [arXiv:1701.03171] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    C. Closset, H. Kim and B. Willett, Seifert fibering operators in 3d N = 2 theories, JHEP11 (2018) 004 [arXiv:1807.02328] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    A. Pittelli, A refined N = 2 chiral multiplet on twisted AdS 2× S 1 , arXiv:1812.11151 [INSPIRE].
  24. [24]
    E. Witten, Analytic continuation of Chern-Simons theory, AMS/IP Stud. Adv. Math.50 (2011) 347 [arXiv:1001.2933] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  25. [25]
    T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys.98 (2011) 225 [arXiv:1006.0977] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
  27. [27]
    E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
  28. [28]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys.325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    Y. Yoshida, Factorization of 4d N = 1 superconformal index, arXiv:1403.0891 [INSPIRE].
  30. [30]
    S. Pasquetti, Holomorphic blocks and the 5d AGT correspondence, J. Phys.A 50 (2017) 443016 [arXiv:1608.02968] [INSPIRE].
  31. [31]
    P. Longhi, F. Nieri and A. Pittelli, Localization of 4d N = 1 theories on D 2× T 2 , arXiv:1906.02051 [INSPIRE].
  32. [32]
    J.-P. Ramis, J. Sauloy and C. Zhang, Local analytic classification of q-difference equations, Astérisque355 (2013) 1 [arXiv:0903.0853].MathSciNetzbMATHGoogle Scholar
  33. [33]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys.B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
  34. [34]
    A. Tabler, Monodromy of q-difference equations in 3D supersymmetric gauge theories, Master thesis, Arnold Sommerfeld Center for Theoretical Physics, Munich, Germany (2017).
  35. [35]
    G.N. Watson, The continuation of functions defined by generalized hypergeometric series, Trans. Cambridge Phil. Soc.21 (1910) 281.Google Scholar
  36. [36]
    T. Morita, A connection formula of the Hahn-Exton q-Bessel function, SIGMA7 (2011) 115 [arXiv:1105.1998].
  37. [37]
    T. Morita, The Stokes phenomenon for the Ramanujan’s q-difference equation and its higher order extension, arXiv:1404.2541.
  38. [38]
    T. Dreyfus and A. Eloy, q-Borel-Laplace summation for q-difference equations with two slopes, J. Diff. Eq. Appl.22 (2016) 1501 [arXiv:1501.02994].
  39. [39]
    Y. Ohyama, q-Stokes phenomenon of a basic hypergeometric series 1 𝜙1 (0; a; q, x), J. Math. Tokushima Univ.50 (2016) 49.Google Scholar
  40. [40]
    Y. Ohyama and C. Zhang, q-Stokes phenomenon on basic hypergeometric series, in 13thSymmetries and Integrability of Difference Equations, Fukuoka, Japan (2018), pg. 35.
  41. [41]
    S. Adachi, The q-Borel sum of divergent basic hypergeometric series r 𝜙s (a; b; q, x), SIGMA15 (2019) 12 [arXiv:1806.05375].zbMATHGoogle Scholar
  42. [42]
    D. Gaiotto, Z. Komargodski and J. Wu, Curious aspects of three-dimensional N = 1 SCFTs, JHEP08 (2018) 004 [arXiv:1804.02018] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institute of Mathematical Sciences, Homi Bhabha National Institute (HBNI)TaramaniIndia
  2. 2.Department of Physics, Indian Institute of Technology MadrasChennaiIndia
  3. 3.Theory Division, Saha Institute of Nuclear PhysicsKolkataIndia
  4. 4.Harish-Chandra Research Institute, Homi Bhabha National Institute (HBNI)AllahabadIndia

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