Abstract
We study \( \mathcal{N}=2 \) superconformal theories with gauge group SU(N ) and 2N fundamental flavours in a locus of the Coulomb branch with a \( {\mathbb{Z}}_N \) symmetry. In this special vacuum, we calculate the prepotential, the dual periods and the period matrix
using equivariant localization. When the flavors are massless, we find that the period matrix is completely specified by \( \left[\frac{N}{2}\right] \) effective couplings. On each of these, we show that the S-duality group acts as a generalized triangle group and that its hauptmodul can be used to write a non-perturbatively exact relation between each effective coupling and the bare one. For N = 2, 3, 4 and 6, the generalized triangle group is an arithmetic Hecke group which contains a subgroup that is also a congruence subgroup of the modular group \( \mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) \). For these cases, we introduce mass deformations that respect the symmetries of the special vacuum and show that the constraints arising from S-duality make it possible to resum the instanton contributions to the period matrix in terms of meromorphic modular forms which solve modular anomaly equations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Teschner, New dualities of supersymmetric gauge theories, Mathematical Physics Studies, Springer (2016).
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [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. 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].
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].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [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].
S.K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, R.R. John and A. Lerda, Non-perturbative studies of N = 2 conformal quiver gauge theories, Fortsch. Phys. 63 (2015) 259 [arXiv:1502.05581] [INSPIRE].
J.A. Minahan, D. Nemeschansky and N.P. 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, A. Lerda and I. Pesando, Modular anomaly equation, heat kernel and S-duality in N = 2 theories, JHEP 11 (2013) 123 [arXiv:1307.6648] [INSPIRE].
M. Billó, M. Frau, L. Gallot, A. Lerda and I. Pesando, Deformed N = 2 theories, generalized recursion relations and S-duality, JHEP 04 (2013) 039 [arXiv:1302.0686] [INSPIRE].
M. Billó et al., Modular anomaly equations in \( \mathcal{N}=2 \) ∗ theories and their large-N limit, JHEP 10 (2014) 131 [arXiv:1406.7255] [INSPIRE].
M. Billò, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential of \( \mathcal{N}=2 \) ⋆ theories (II): the non-simply laced algebras, JHEP 11 (2015) 026 [arXiv:1507.08027] [INSPIRE].
M. Billò, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential in \( \mathcal{N}=2 \) ⋆ theories (I): the ADE algebras, JHEP 11 (2015) 024 [arXiv:1507.07709] [INSPIRE].
M. Billò, M. Frau, F. Fucito, A. Lerda and J.F. Morales, Resumming instantons in N = 2∗ theories with arbitrary gauge groups, arXiv:1602.00273 [INSPIRE].
S.K. Ashok, M. Billò, E. Dell’Aquila, M. Frau, A. Lerda and M. Raman, Modular anomaly equations and S-duality in \( \mathcal{N}=2 \) conformal SQCD, JHEP 10 (2015) 091 [arXiv:1507.07476] [INSPIRE].
P.C. Argyres and S. Pelland, Comparing instanton contributions with exact results in N = 2 supersymmetric scale invariant theories, JHEP 03 (2000) 014 [hep-th/9911255] [INSPIRE].
C.F. Doran, T. Gannon, H. Movasati and K.M. Shokri, Automorphic forms for triangle groups, Commun. Num. Theor Phys. 07 (2013) 689 [arXiv:1307.4372] [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [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. 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].
J.A. Minahan and D. Nemeschansky, Hyperelliptic curves for supersymmetric Yang-Mills, Nucl. Phys. B 464 (1996) 3 [hep-th/9507032] [INSPIRE].
J.A. Minahan and D. Nemeschansky, N = 2 super Yang-Mills and subgroups of \( \mathrm{S}\mathrm{L}\left(2,\;\mathbb{Z}\right) \), Nucl. Phys. B 468 (1996) 72 [hep-th/9601059] [INSPIRE].
J.A. Minahan, Duality symmetries for N = 2 supersymmetric QCD with vanishing β-functions, Nucl. Phys. B 537 (1999) 243 [hep-th/9806246] [INSPIRE].
E. D’Hoker and D.H. Phong, Lectures on supersymmetric Yang-Mills theory and integrable systems, hep-th/9912271 [INSPIRE].
M. Billó et al., Non-perturbative gauge/gravity correspondence in N = 2 theories, JHEP 08 (2012) 166 [arXiv:1206.3914] [INSPIRE].
O. Aharony and S. Yankielowicz, Exact electric-magnetic duality in N = 2 supersymmetric QCD theories, Nucl. Phys. B 473 (1996) 93 [hep-th/9601011] [INSPIRE].
P.C. Argyres and A. Buchel, The nonperturbative gauge coupling of N = 2 supersymmetric theories, Phys. Lett. B 442 (1998) 180 [hep-th/9806234] [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].
T.W. Grimm, A. Klemm, M. Mariño and M. Weiss, Direct integration of the topological string, JHEP 08 (2007) 058 [hep-th/0702187] [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].
T. Gannon, The algebraic meaning of being a hauptmodul, in Moonshine: the first quarter century and beyond, J. Lepowsky et al. eds., Cambridge University Press, Cambridge U.K. (2010).
D. Zagier, Traces of singular moduli, in Motives, polylogarithms and hodge theory. Part I, L.K.F. Bogomolov ed., International Press, Somerville U.S.A. (2002).
P.C. Argyres, M.R. Plesser and A.D. Shapere, The Coulomb phase of N = 2 supersymmetric QCD, Phys. Rev. Lett. 75 (1995) 1699 [hep-th/9505100] [INSPIRE].
P.C. Argyres, S duality and global symmetries in N = 2 supersymmetric field theory, Adv. Theor. Math. Phys. 2 (1998) 293 [hep-th/9706095] [INSPIRE].
N. Koblitz, Introduction to elliptic curves and modular forms, 2nd edition, Springer, Germany (1993).
T.M. Apostol, Modular functions and Dirichlet series in number theory, 2nd edition, Springer, Germany (1990).
H.H. Chan and S. Cooper, Rational analogues of Ramanujan’s series for 1/π, Math. Proc. Cambridge Phil. Soc. 153 (2012) 361.
N. Wyllard, A N −1 conformal Toda field theory correlation functions from conformal N = 2 SU(N ) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
A.-K. Kashani-Poor and J. Troost, The toroidal block and the genus expansion, JHEP 03 (2013) 133 [arXiv:1212.0722] [INSPIRE].
A.-K. Kashani-Poor and J. Troost, Transformations of spherical blocks, JHEP 10 (2013) 009 [arXiv:1305.7408] [INSPIRE].
A.-K. Kashani-Poor and J. Troost, Quantum geometry from the toroidal block, JHEP 08 (2014) 117 [arXiv:1404.7378] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, Zamolodchikov asymptotic formula and instanton expansion in N = 2 SUSY N f = 2N c QCD, JHEP 11 (2009) 048 [arXiv:0909.3338] [INSPIRE].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [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: 1601.01827
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
Ashok, S.K., Dell’Aquila, E., Lerda, A. et al. S-duality, triangle groups and modular anomalies in \( \mathcal{N}=2 \) SQCD. J. High Energ. Phys. 2016, 118 (2016). https://doi.org/10.1007/JHEP04(2016)118
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2016)118