Abstract
We study \( \mathcal{N}=2 \) SU(2) supersymmetric QCD with massive hypermultiplets deformed in the Nekrasov-Shatashvili limit of the Omega-background. The prepotential of the low-energy effective theory is determined by the WKB solution of the quantum Seiberg-Witten curve. We calculate the deformed Seiberg-Witten periods around the massless monoplole point explicitly up to the fourth order in the deformation parameter.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
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].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
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].
D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, J. Phys. Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
M.-X. Huang and A. Klemm, Holomorphicity and modularity in Seiberg-Witten theories with matter, JHEP 07 (2010) 083 [arXiv:0902.1325] [INSPIRE].
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].
K. Maruyoshi and M. Taki, Deformed prepotential, quantum integrable system and Liouville field theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].
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].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].
R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].
A. Mironov and A. Morozov, Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [INSPIRE].
Y. Zenkevich, Nekrasov prepotential with fundamental matter from the quantum spin chain, Phys. Lett. B 701 (2011) 630 [arXiv:1103.4843] [INSPIRE].
M. Beccaria, On the large Ω-deformations in the Nekrasov-Shatashvili limit of N = 2∗ SYM, JHEP 07 (2016) 055 [arXiv:1605.00077] [INSPIRE].
W. He, A new treatment for some periodic Schrödinger operators II: the wave function, arXiv:1608.05350 [INSPIRE].
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].
A. Popolitov, On relation between Nekrasov functions and BS periods in pure SU(N) case, arXiv:1001.1407 [INSPIRE].
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].
D. Krefl, Non-perturbative quantum geometry II, JHEP 12 (2014) 118 [arXiv:1410.7116] [INSPIRE].
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].
A.-K. Kashani-Poor and J. Troost, Pure N = 2 super Yang-Mills and exact WKB, JHEP 08 (2015) 160 [arXiv:1504.08324] [INSPIRE].
S.K. Ashok, D.P. Jatkar, R.R. John, M. Raman and J. Troost, Exact WKB analysis of N =2 gauge theories, JHEP 07(2016) 115[arXiv:1604.05520] [INSPIRE].
G. Basar, G.V. Dunne and M. Ünsal, Quantum geometry of resurgent perturbative/nonperturbative relations, JHEP 05 (2017) 087 [arXiv:1701.06572] [INSPIRE].
N. Dorey, V.V. Khoze and M.P. Mattis, On N = 2 supersymmetric QCD with four flavors, Nucl. Phys. B 492 (1997) 607 [hep-th/9611016] [INSPIRE].
A. Hanany and Y. Oz, On the quantum moduli space of vacua of N = 2 supersymmetric SU(N c ) gauge theories, Nucl. Phys. B 452 (1995) 283 [hep-th/9505075] [INSPIRE].
A. Ceresole, R. D’Auria and S. Ferrara, On the geometry of moduli space of vacua in N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 339 (1994) 71 [hep-th/9408036] [INSPIRE].
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].
K. Ito and S.-K. Yang, Prepotentials in N = 2 SU(2) supersymmetric Yang-Mills theory with massless hypermultiplets, Phys. Lett. B 366 (1996) 165 [hep-th/9507144] [INSPIRE].
Y. Ohta, Prepotential of N = 2 SU(2) Yang-Mills gauge theory coupled with a massive matter multiplet, J. Math. Phys. 37 (1996) 6074 [hep-th/9604051] [INSPIRE].
Y. Ohta, Prepotentials of N = 2 SU(2) Yang-Mills theories coupled with massive matter multiplets, J. Math. Phys. 38 (1997) 682 [hep-th/9604059] [INSPIRE].
T. Masuda and H. Suzuki, Periods and prepotential of N = 2 SU(2) supersymmetric Yang-Mills theory with massive hypermultiplets, Int. J. Mod. Phys. A 12 (1997) 3413 [hep-th/9609066] [INSPIRE].
A. Erdelyi et al., Higher transcendental functions, volume 1, MacGraw-Hill, New-York U.S.A., (1953).
M.-X. Huang, On gauge theory and topological string in Nekrasov-Shatashvili limit, JHEP 06 (2012) 152 [arXiv:1205.3652] [INSPIRE].
W. He, N = 2 supersymmetric QCD and elliptic potentials, JHEP 11 (2014) 030 [arXiv:1306.4590] [INSPIRE].
M. Piatek, Classical conformal blocks from TBA for the elliptic Calogero-Moser system, JHEP 06 (2011) 050 [arXiv:1102.5403] [INSPIRE].
F. Ferrari and M. Piatek, Liouville theory, N = 2 gauge theories and accessory parameters, JHEP 05 (2012) 025 [arXiv:1202.2149] [INSPIRE].
M. Piatek and A.R. Pietrykowski, Classical irregular blocks, Hill’s equation and PT-symmetric periodic complex potentials, JHEP 07 (2016) 131 [arXiv:1604.03574] [INSPIRE].
Y. Ohta, Differential equations for scaling relation in N = 2 supersymmetric SU(2) Yang-Mills theory coupled with massive hypermultiplet, J. Math. Phys. 40 (1999) 1891 [hep-th/9809180] [INSPIRE].
T. Eguchi, K. Hori, K. Ito and S.-K. Yang, Study of N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 471 (1996) 430 [hep-th/9603002] [INSPIRE].
T. Masuda and H. Suzuki, On explicit evaluations around the conformal point in N = 2 supersymmetric Yang-Mills theories, Nucl. Phys. B 495 (1997) 149 [hep-th/9612240] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1705.09120
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ito, K., Kanno, S. & Okubo, T. Quantum periods and prepotential in \( \mathcal{N}=2 \) SU(2) SQCD. J. High Energ. Phys. 2017, 65 (2017). https://doi.org/10.1007/JHEP08(2017)065
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2017)065