Abstract
In this paper we investigate different ways of deriving the A-cycle period as a series in instanton counting parameter q for \( \mathcal{N} \) = 2 SYM with up to four antifundamental hypermultiplets in NS limit of Ω background. We propose a new recursive method for calculating the period and demonstrate its efficiency by explicit calculations. The new way of doing instanton counting is more advantageous compared to known standard techniques and allows to reach substantially higher order terms with less effort. This approach is applied for the pure case as well as for the case with several hypermultiplets.
In addition we suggest a numerical method for deriving the A-cycle period for arbitrary values of q. In the case when one has no hypermultiplets for the A-cycle an analytic expression for large q asymptotics is obtained using a conjecture by Alexei Zamolodchikov. We demonstrate that this expression is in convincing agreement with the numerical approach.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, NATO Sci. Ser. C 520 (1999) 359 [hep-th/9801061] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
R. Flume and R. Poghossian, An Algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hep-th/0208176] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in 16th International Congress on Mathematical Physics, 8, 2009, DOI [arXiv:0908.4052] [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].
N. Seiberg, Notes on quantum Liouville theory and quantum gravity, Prog. Theor. Phys. Suppl. 102 (1990) 319 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
A.B. Zamolodchikov, conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].
R. Poghossian, Recurrence relations for the \( {\mathcal{W}}_3 \) conformal blocks and \( \mathcal{N} \) = 2 SYM partition functions, JHEP 11 (2017) 053 [Erratum ibid. 01 (2018) 088] [arXiv:1705.00629] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursion representation of the Neveu-Schwarz superconformal block, JHEP 03 (2007) 032 [hep-th/0611266] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Elliptic recurrence representation of the N = 1 superconformal blocks in the Ramond sector, JHEP 11 (2008) 060 [arXiv:0810.1203] [INSPIRE].
S. Alekseev, A. Gorsky and M. Litvinov, Toward the Pole, JHEP 03 (2020) 157 [arXiv:1911.01334] [INSPIRE].
M. Beccaria, On the large Ω-deformations in the Nekrasov-Shatashvili limit of \( \mathcal{N} \) = 2* SYM, JHEP 07 (2016) 055 [arXiv:1605.00077] [INSPIRE].
A. Gorsky, A. Milekhin and N. Sopenko, Bands and gaps in Nekrasov partition function, JHEP 01 (2018) 133 [arXiv:1712.02936] [INSPIRE].
R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].
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. 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].
M.-x. Huang, A.-K. Kashani-Poor and A. Klemm, The Ω deformed B-model for rigid \( \mathcal{N} \) = 2 theories, Annales Henri Poincaré 14 (2013) 425 [arXiv:1109.5728] [INSPIRE].
M.-x. Huang, A. Klemm, J. Reuter and M. Schiereck, Quantum geometry of del Pezzo surfaces in the Nekrasov-Shatashvili limit, JHEP 02 (2015) 031 [arXiv:1401.4723] [INSPIRE].
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].
A. Mironov and A. Morozov, Nekrasov Functions and Exact Bohr-Zommerfeld Integrals, JHEP 04 (2010) 040 [arXiv:0910.5670] [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].
K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].
G. Wolf, Mathieu Functions and Hill’s Equation NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/28.
A. Zamolodchikov, Generalized Mathieu equation and Liouville TBA, 2000, in Quantum Field Theories in Two Dimensions. Vol. 2, World Scientific, New York U.S.A. (2012).
D. Fioravanti, H. Poghosyan and R. Poghossian, T, Q and periods in SU(3) \( \mathcal{N} \) = 2 SYM, JHEP 03 (2020) 049 [arXiv:1909.11100] [INSPIRE].
N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP 03 (2016) 181 [arXiv:1512.05388] [INSPIRE].
G. Poghosyan and R. Poghossian, VEV of Baxter’s Q-operator in N = 2 gauge theory and the BPZ differential equation, JHEP 11 (2016) 058 [arXiv:1602.02772] [INSPIRE].
G. Poghosyan, VEV of Q-operator in U(1) linear quiver 5d gauge theories, arXiv:1801.04303 [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, Commun. Math. Phys. 357 (2018) 519 [arXiv:1312.6689] [INSPIRE].
P. Dorey and R. Tateo, Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations, J. Phys. A 32 (1999) L419 [hep-th/9812211] [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Spectral determinants for Schrödinger equation and Q operators of conformal field theory, J. Statist. Phys. 102 (2001) 567 [hep-th/9812247] [INSPIRE].
P. Dorey, C. Dunning and R. Tateo, The ODE/IM Correspondence, J. Phys. A 40 (2007) R205 [hep-th/0703066] [INSPIRE].
A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equation in sinh-Gordon model, J. Phys. A 39 (2006) 12863 [hep-th/0005181] [INSPIRE].
M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett. B 357 (1995) 342 [hep-th/9506102] [INSPIRE].
R. Flume, F. Fucito, J.F. Morales and R. Poghossian, Matone’s relation in the presence of gravitational couplings, JHEP 04 (2004) 008 [hep-th/0403057] [INSPIRE].
D. Fioravanti and D. Gregori, Integrability and cycles of deformed \( \mathcal{N} \) = 2 gauge theory, Phys. Lett. B 804 (2020) 135376 [arXiv:1908.08030] [INSPIRE].
A. Grassi, J. Gu and M. Mariño, Non-perturbative approaches to the quantum Seiberg-Witten curve, JHEP 07 (2020) 106 [arXiv:1908.07065] [INSPIRE].
R. Poghossian, Deformed SW curve and the null vector decoupling equation in Toda field theory, JHEP 04 (2016) 070 [arXiv:1601.05096] [INSPIRE].
S.K. Ashok, D.P. Jatkar, R.R. John, M. Raman and J. Troost, Exact WKB analysis of \( \mathcal{N} \) = 2 gauge theories, JHEP 07 (2016) 115 [arXiv:1604.05520] [INSPIRE].
A. Litvinov, S. Lukyanov, N. Nekrasov and A. Zamolodchikov, Classical Conformal Blocks and Painleve VI, JHEP 07 (2014) 144 [arXiv:1309.4700] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2010.08498
Rights and permissions
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.
About this article
Cite this article
Poghosyan, H. Recursion relation for instanton counting for SU(2) \( \mathcal{N} \) = 2 SYM in NS limit of Ω background. J. High Energ. Phys. 2021, 88 (2021). https://doi.org/10.1007/JHEP05(2021)088
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2021)088