Abstract
We use localization techniques to study the non-perturbative properties of an \( \mathcal{N}=2 \) superconformal gauge theory with gauge group SU(3) and six fundamental flavours. The instanton corrections to the prepotential, the dual periods and the period matrix are calculated in a locus of special vacua possessing a ℤ 3 symmetry. In a semiclassical expansion, we show that these observables are constrained by S-duality via a modular anomaly equation which takes the form of a recursion relation. The solutions of the recursion relation are quasi-modular functions of Γ1 (3), which is a subgroup of the S-duality group and is also a congruence subgroup of SL(2, ℤ).
Article PDF
Similar content being viewed by others
References
J. Teschner, Exact results on N = 2 supersymmetric gauge theories, arXiv:1412.7145 [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. Billò, 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. Billò, 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.-X. Huang, Modular anomaly from holomorphic anomaly in mass deformed N = 2 superconformal field theories, Phys. Rev. D 87 (2013) 105010 [arXiv:1302.6095] [INSPIRE].
M. Billò et al., Modular anomaly equations in 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 N = 2∗ theories (I): the ADE algebras,arXiv:1507.07709[INSPIRE].
M. Billò, M. Frau, F. Fucito, A. Lerda and J.F. Morales, S-duality and the prepotential in N = 2∗ theories (II): the non-simply laced algebras,arXiv:1507.08027[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].
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].
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. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [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].
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].
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 SL (2, Z), Nucl. Phys. B 468 (1996) 72 [hep-th/9601059] [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].
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. Koblitz, Introduction to elliptic curves and modular forms, 2nd edition, Springer-Verlag, Germany (1993).
T.M. Apostol, Modular functions and Dirichlet series in number theory, 2nd edition, Springer-Verlag, Germany (1990).
M. Billò et al., Non-perturbative gauge/gravity correspondence in N = 2 theories, JHEP 08 (2012) 166 [arXiv:1206.3914] [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].
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].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E 6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].
J.A. Minahan and D. Nemeschansky, Superconformal fixed points with E n global symmetry, Nucl. Phys. B 489 (1997) 24 [hep-th/9610076] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [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. 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].
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: 1507.07476
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., Billò, M., Dell’Aquila, E. et al. Modular anomaly equations and S-duality in \( \mathcal{N}=2 \) conformal SQCD. J. High Energ. Phys. 2015, 91 (2015). https://doi.org/10.1007/JHEP10(2015)091
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2015)091