Abstract
We consider the exact coupling constant dependence of extremal correlation functions of \( \mathcal{N}=2 \) chiral primary operators in 4d \( \mathcal{N}=2 \) superconformal gauge theories with gauge group SU(N) and N f = 2N massless fundamental hypermultiplets. The 2- and 3-point functions, viewed as functions of the exactly marginal coupling constant and theta angle, obey the tt * equations. In the case at hand, the tt * equations form a set of complicated non-linear coupled matrix equations. We point out that there is an ad hoc self-consistent ansatz that reduces this set of partial differential equations to a sequence of decoupled semi-infinite Toda chains, similar to the one encountered previously in the special case of SU(2) gauge group. This ansatz requires a surprising new non-renormalization theorem in \( \mathcal{N}=2 \) superconformal field theories. We derive a general 3-loop perturbative formula for 2- and 3-point functions in the \( \mathcal{N}=2 \) chiral ring of the SU(N) theory, and in all explicitly computed examples we find agreement with the tt * equations, as well as the above-mentioned ansatz. This is suggestive evidence for an interesting non-perturbative conjecture about the structure of the \( \mathcal{N}=2 \) chiral ring in this class of theories. We discuss several implications of this conjecture. For example, it implies that the holonomy of the vector bundles of chiral primaries over the superconformal manifold is reducible. It also implies that a specific subset of extremal correlation functions can be computed in the SU(N) theory using information solely from the S 4 partition function of the theory obtained by supersymmetric localization.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].
K. Papadodimas, Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE].
W. Lerche, C. Vafa and N.P. Warner, Chiral Rings in N = 2 Superconformal Theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, tt * equations, localization and exact chiral rings in 4d \( \mathcal{N}=2 \) SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
S. Cecotti and C. Vafa, Exact results for supersymmetric σ-models, Phys. Rev. Lett. 68 (1992) 903 [hep-th/9111016] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, Exact correlation functions in SU(2) \( \mathcal{N}=2 \) superconformal QCD, Phys. Rev. Lett. 113 (2014) 251601 [arXiv:1409.4217] [INSPIRE].
E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere Partition Functions and the Zamolodchikov Metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].
J. Gomis and N. Ishtiaque, Kähler potential and ambiguities in 4d \( \mathcal{N}=2 \) SCFTs, JHEP 04 (2015) 169 [arXiv:1409.5325] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
T.W. Brown, R. de Mello Koch, S. Ramgoolam and N. Toumbas, Correlators, Probabilities and Topologies in N = 4 SYM, JHEP 03 (2007) 072 [hep-th/0611290] [INSPIRE].
S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large-N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].
E. D’Hoker, D.Z. Freedman and W. Skiba, Field theory tests for correlators in the AdS/CFT correspondence, Phys. Rev. D 59 (1999) 045008 [hep-th/9807098] [INSPIRE].
E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Extremal correlators in the AdS/CFT correspondence, hep-th/9908160 [INSPIRE].
K.A. Intriligator, Bonus symmetries of N = 4 super Yang-Mills correlation functions via AdS duality, Nucl. Phys. B 551 (1999) 575 [hep-th/9811047] [INSPIRE].
K.A. Intriligator and W. Skiba, Bonus symmetry and the operator product expansion of N = 4 Super Yang-Mills, Nucl. Phys. B 559 (1999) 165 [hep-th/9905020] [INSPIRE].
B. Eden, P.S. Howe and P.C. West, Nilpotent invariants in N = 4 SYM, Phys. Lett. B 463 (1999) 19 [hep-th/9905085] [INSPIRE].
A. Petkou and K. Skenderis, A nonrenormalization theorem for conformal anomalies, Nucl. Phys. B 561 (1999) 100 [hep-th/9906030] [INSPIRE].
P.S. Howe, C. Schubert, E. Sokatchev and P.C. West, Explicit construction of nilpotent covariants in N = 4 SYM, Nucl. Phys. B 571 (2000) 71 [hep-th/9910011] [INSPIRE].
P.J. Heslop and P.S. Howe, OPEs and three-point correlators of protected operators in N = 4 SYM, Nucl. Phys. B 626 (2002) 265 [hep-th/0107212] [INSPIRE].
M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE].
J. Louis, H. Triendl and M. Zagermann, \( \mathcal{N}=4 \) supersymmetric AdS 5 vacua and their moduli spaces, JHEP 10 (2015) 083 [arXiv:1507.01623] [INSPIRE].
D. Binosi and L. Theussl, JaxoDraw: A graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun. 161 (2004) 76 [hep-ph/0309015] [INSPIRE].
D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: A graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun. 180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].
S. Penati, A. Santambrogio and D. Zanon, More on correlators and contact terms in N = 4 SYM at order g 4, Nucl. Phys. B 593 (2001) 651 [hep-th/0005223] [INSPIRE].
R. Andree and D. Young, Wilson Loops in N = 2 Superconformal Yang-Mills Theory, JHEP 09 (2010) 095 [arXiv:1007.4923] [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: 1508.03077
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
Baggio, M., Niarchos, V. & Papadodimas, K. On exact correlation functions in SU(N) \( \mathcal{N}=2 \) superconformal QCD. J. High Energ. Phys. 2015, 198 (2015). https://doi.org/10.1007/JHEP11(2015)198
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2015)198