Abstract
We study the non-perturbative properties of \( \mathcal{N}=2 \) super conformal field theories in four dimensions using localization techniques. In particular we consider SU(2) gauge theories, deformed by a generic ϵ-background, with four fundamental flavors or with one adjoint hypermultiplet. In both cases we explicitly compute the first few instanton corrections to the partition function and the prepotential using Nekrasov’s approach. These results allow us to reconstruct exact expressions involving quasi-modular functions of the bare gauge coupling constant and to show that the prepotential terms satisfy a modular anomaly equation that takes the form of a recursion relation with an explicitly ϵ-dependent term. We then investigate the implications of this recursion relation on the modular properties of the effective theory and find that with a suitable redefinition of the prepotential and of the effective coupling it is possible, at least up to the third order in the deformation parameters, to cast the S-duality relations in the same form as they appear in the Seiberg-Witten solution of the undeformed theory.
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].
E. D’Hoker and D. Phong, Lectures on supersymmetric Yang-Mills theory and integrable systems, hep-th/9912271 [INSPIRE].
J. Minahan, D. Nemeschansky and N. Warner, Instanton expansions for mass deformed N = 4 super Yang-Mills theories, Nucl. Phys. B 528 (1998) 109 [hep-th/9710146] [INSPIRE].
M. Billó, M. Frau, L. Gallot and A. Lerda, The exact 8d chiral ring from 4d recursion relations, JHEP 11 (2011) 077 [arXiv:1107.3691] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].
A. Lossev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061 [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, D particle bound states and generalized instantons, Commun. Math. Phys. 209 (2000) 77 [hep-th/9803265] [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].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [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].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].
M. Mariño and N. Wyllard, A note on instanton counting for N = 2 gauge theories with classical gauge groups, JHEP 05 (2004) 021 [hep-th/0404125] [INSPIRE].
M. Billó et al., Exotic instanton counting and heterotic/type-I-prime duality, JHEP 07 (2009) 092 [arXiv:0905.4586] [INSPIRE].
F. Fucito, J.F. Morales and R. Poghossian, Exotic prepotentials from D(−1)D7 dynamics, JHEP 10 (2009) 041 [arXiv:0906.3802] [INSPIRE].
M. Billó et al., Stringy instanton corrections to N = 2 gauge couplings, JHEP 05 (2010) 107 [arXiv:1002.4322] [INSPIRE].
H. Nakajima and K. Yoshioka, Lectures on instanton counting, math.AG/0311058 [INSPIRE].
M. Billó, M. Frau, F. Fucito and A. Lerda, Instanton calculus in RR background and the topological string, JHEP 11 (2006) 012 [hep-th/0606013] [INSPIRE].
K. Ito, H. Nakajima, T. Saka and S. Sasaki, N = 2 instanton effective action in Ω-background and D3/D(−1)-brane system in RR background, JHEP 11 (2010) 093 [arXiv:1009.1212] [INSPIRE].
I. Antoniadis, E. Gava, K. Narain and T. Taylor, Topological amplitudes in string theory, Nucl. Phys. B 413 (1994) 162 [hep-th/9307158] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993) 279 [hep-th/9302103] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].
A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [INSPIRE].
M.-X. Huang and A. Klemm, Holomorphicity and modularity in Seiberg-Witten theories with matter, JHEP 07 (2010) 083 [arXiv:0902.1325] [INSPIRE].
I. Antoniadis, S. Hohenegger, K. Narain and T. Taylor, Deformed topological partition function and Nekrasov backgrounds, Nucl. Phys. B 838 (2010) 253 [arXiv:1003.2832] [INSPIRE].
D. Krefl and J. Walcher, Extended holomorphic anomaly in gauge theory, Lett. Math. Phys. 95 (2011) 67 [arXiv:1007.0263] [INSPIRE].
M.-X. Huang and A. Klemm, Direct integration for general Ω backgrounds, Adv. Theor. Math. Phys. 16 (2012), no. 3 805–849 [arXiv:1009.1126] [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].
F. Fucito, J. 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].
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, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
A. Mironov, A. Morozov and S. Shakirov, Matrix model conjecture for exact BS periods and Nekrasov functions, JHEP 02 (2010) 030 [arXiv:0911.5721] [INSPIRE].
M.-X. Huang, A.-K. Kashani-Poor and A. Klemm, The Ω deformed B-model for rigid N = 2 theories, Annales Henri Poincaré 14 (2013) 425 [arXiv:1109.5728] [INSPIRE].
M.-X. Huang, On gauge theory and topological string in Nekrasov-Shatashvili limit, JHEP 06 (2012) 152 [arXiv:1205.3652] [INSPIRE].
A.-K. Kashani-Poor and J. Troost, The toroidal block and the genus expansion, JHEP 03 (2013) 133 [arXiv:1212.0722] [INSPIRE].
D. Galakhov, A. Mironov and A. Morozov, S-duality as a beta-deformed Fourier transform, JHEP 08 (2012) 067 [arXiv:1205.4998] [INSPIRE].
M. Billó et al., Non-perturbative gauge/gravity correspondence in N = 2 theories, JHEP 08 (2012) 166 [arXiv:1206.3914] [INSPIRE].
T. Okuda and V. Pestun, On the instantons and the hypermultiplet mass of N = 2* super Yang-Mills on S 4, JHEP 03 (2012) 017 [arXiv:1004.1222] [INSPIRE].
F. Fucito, J.F. Morales and R. Poghossian, Multi instanton calculus on ALE spaces, Nucl. Phys. B 703 (2004) 518 [hep-th/0406243] [INSPIRE].
J. Minahan, D. Nemeschansky and N. Warner, Partition functions for BPS states of the noncritical E 8 string, Adv. Theor. Math. Phys. 1 (1998) 167 [hep-th/9707149] [INSPIRE].
T.W. Grimm, A. Klemm, M. Mariño and M. Weiss, Direct integration of the topological string, JHEP 08 (2007) 058 [hep-th/0702187] [INSPIRE].
M. Billó, L. Gallot, A. Lerda and I. Pesando, F-theoretic versus microscopic description of a conformal N = 2 SYM theory, JHEP 11 (2010) 041 [arXiv:1008.5240] [INSPIRE].
T. Dimofte and S. Gukov, Chern-Simons theory and S-duality, arXiv:1106.4550 [INSPIRE].
E. Witten, Quantum background independence in string theory, hep-th/9306122 [INSPIRE].
M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, work in progress.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1302.0686
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Billó, M., Frau, M., Gallot, L. et al. Deformed \( \mathcal{N}=2 \) theories, generalized recursion relations and S-duality. J. High Energ. Phys. 2013, 39 (2013). https://doi.org/10.1007/JHEP04(2013)039
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2013)039