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.
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References
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett.B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
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].
N. Dorey and D. Tong, Mirror symmetry and toric geometry in three-dimensional gauge theories, JHEP05 (2000) 018 [hep-th/9911094] [INSPIRE].
D. Tong, Dynamics of N = 2 supersymmetric Chern-Simons theories, JHEP07 (2000) 019 [hep-th/0005186] [INSPIRE].
E. Witten, Topological quantum field theory, Commun. Math. Phys.117 (1988) 353 [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys.313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
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].
D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
Y. Imamura and D. Yokoyama, N = 2 supersymmetric theories on squashed three-sphere, Phys. Rev.D 85 (2012) 025015 [arXiv:1109.4734] [INSPIRE].
J. Nian, Localization of supersymmetric Chern-Simons-matter theory on a squashed S3with SU(2) × U(1) isometry, JHEP07 (2014) 126 [arXiv:1309.3266] [INSPIRE].
V. Pestun and M. Zabzine, Introduction to localization in quantum field theory, J. Phys.A 50 (2017) 443001 [arXiv:1608.02953] [INSPIRE].
S. Pasquetti, Factorisation of N = 2 theories on the squashed 3-sphere, JHEP04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys.17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic blocks in three dimensions, JHEP12 (2014) 177 [arXiv:1211.1986] [INSPIRE].
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].
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].
Y. Imamura, H. Matsuno and D. Yokoyama, Factorization of the S3/Znpartition function, Phys. Rev.D 89 (2014) 085003 [arXiv:1311.2371] [INSPIRE].
F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4d, JHEP11 (2015) 155 [arXiv:1507.00261] [INSPIRE].
C. Closset, H. Kim and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP03 (2017) 074 [arXiv:1701.03171] [INSPIRE].
C. Closset, H. Kim and B. Willett, Seifert fibering operators in 3d N = 2 theories, JHEP11 (2018) 004 [arXiv:1807.02328] [INSPIRE].
A. Pittelli, A refined N = 2 chiral multiplet on twisted AdS2× S1 , arXiv:1812.11151 [INSPIRE].
E. Witten, Analytic continuation of Chern-Simons theory, AMS/IP Stud. Adv. Math.50 (2011) 347 [arXiv:1001.2933] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys.98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
E. Witten, A new look at the path integral of quantum mechanics, arXiv:1009.6032 [INSPIRE].
E. Witten, Fivebranes and knots, arXiv:1101.3216 [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys.325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
Y. Yoshida, Factorization of 4d N = 1 superconformal index, arXiv:1403.0891 [INSPIRE].
S. Pasquetti, Holomorphic blocks and the 5d AGT correspondence, J. Phys.A 50 (2017) 443016 [arXiv:1608.02968] [INSPIRE].
P. Longhi, F. Nieri and A. Pittelli, Localization of 4d N = 1 theories on D2× T2 , arXiv:1906.02051 [INSPIRE].
J.-P. Ramis, J. Sauloy and C. Zhang, Local analytic classification of q-difference equations, Astérisque355 (2013) 1 [arXiv:0903.0853].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys.B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
A. Tabler, Monodromy of q-difference equations in 3D supersymmetric gauge theories, Master thesis, Arnold Sommerfeld Center for Theoretical Physics, Munich, Germany (2017).
G.N. Watson, The continuation of functions defined by generalized hypergeometric series, Trans. Cambridge Phil. Soc.21 (1910) 281.
T. Morita, A connection formula of the Hahn-Exton q-Bessel function, SIGMA7 (2011) 115 [arXiv:1105.1998].
T. Morita, The Stokes phenomenon for the Ramanujan’s q-difference equation and its higher order extension, arXiv:1404.2541.
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].
Y. Ohyama, q-Stokes phenomenon of a basic hypergeometric series1 𝜙1 (0; a; q, x), J. Math. Tokushima Univ.50 (2016) 49.
Y. Ohyama and C. Zhang, q-Stokes phenomenon on basic hypergeometric series, in 13thSymmetries and Integrability of Difference Equations, Fukuoka, Japan (2018), pg. 35.
S. Adachi, The q-Borel sum of divergent basic hypergeometric seriesr 𝜙s (a; b; q, x), SIGMA15 (2019) 12 [arXiv:1806.05375].
D. Gaiotto, Z. Komargodski and J. Wu, Curious aspects of three-dimensional N = 1 SCFTs, JHEP08 (2018) 004 [arXiv:1804.02018] [INSPIRE].
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ArXiv ePrint: 1907.05031
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Ashok, S.K., Subramanian, P.N.B., Bawane, A. et al. Exact WKB analysis of ℂℙ1 holomorphic blocks. J. High Energ. Phys. 2019, 75 (2019). https://doi.org/10.1007/JHEP10(2019)075
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DOI: https://doi.org/10.1007/JHEP10(2019)075