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Journal of High Energy Physics

, 2019:11 | Cite as

Universality of Toda equation in \( \mathcal{N}=2 \) superconformal field theories

  • Antoine Bourget
  • Diego Rodriguez-GomezEmail author
  • Jorge G. Russo
Open Access
Regular Article - Theoretical Physics
  • 5 Downloads

Abstract

We show that extremal correlators in all Lagrangian \( \mathcal{N}=2 \) superconformal field theories with a simple gauge group, when suitably defined the \( {\mathbb{S}}^4 \), are governed by the same universal Toda system of equations, which dictates the structure of extremal correlators to all orders in the perturbation series. A key point is the construction of a convenient orthogonal basis for the chiral ring, by arranging towers of operators in order of increasing dimension, which has the property that the associated two-point functions satisfy decoupled Toda chain equations. We explicitly verify this in all known SCFTs based on SU(N) gauge groups as well as in superconformal QCD based on orthogonal and symplectic groups. As a by-product, we find a surprising non-renormalization property for the \( \mathcal{N}=2 \) SU(N) SCFT with one hypermultiplet in the rank-2 symmetric representation and one hypermultiplet in the rank-2 antisymmetric representation, where the two-loop terms of a large class of supersymmetric observables identically vanish.

Keywords

Conformal Field Theory Supersymmetric Gauge Theory 

Notes

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.

References

  1. [1]
    K. Papadodimas, Topological anti-topological fusion in four-dimensional superconformal field theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. Baggio, V. Niarchos and K. Papadodimas, tt equations, localization and exact chiral rings in 4d N = 2 SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
  3. [3]
    M. Baggio, V. Niarchos and K. Papadodimas, Exact correlation functions in SU(2) N = 2 superconformal QCD, Phys. Rev. Lett. 113 (2014) 251601 [arXiv:1409.4217] [INSPIRE].
  4. [4]
    M. Baggio, V. Niarchos and K. Papadodimas, On exact correlation functions in SU(N) N = 2 superconformal QCD, JHEP 11 (2015) 198 [arXiv:1508.03077] [INSPIRE].
  5. [5]
    E. Gerchkovitz, J. Gomis, N. Ishtiaque, A. Karasik, Z. Komargodski and S.S. Pufu, Correlation functions of Coulomb branch operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. Bourget, A. Pini and D. Rodríguez-Gómez, The importance of being disconnected, a principal extension for serious groups, arXiv:1804.01108 [INSPIRE].
  7. [7]
    P.C. Argyres and M. Martone, Coulomb branches with complex singularities, JHEP 06 (2018) 045 [arXiv:1804.03152] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    T. Bourton, A. Pini and E. Pomoni, 4d N = 3 indices via discrete gauging, JHEP 10 (2018) 131 [arXiv:1804.05396] [INSPIRE].
  9. [9]
    E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere partition functions and the Zamolodchikov metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    D. Rodriguez-Gomez and J.G. Russo, Large N correlation functions in superconformal field theories, JHEP 06 (2016) 109 [arXiv:1604.07416] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    D. Rodriguez-Gomez and J.G. Russo, Operator mixing in large N superconformal field theories on S 4 and correlators with Wilson loops, JHEP 12 (2016) 120 [arXiv:1607.07878] [INSPIRE].
  13. [13]
    M. Baggio, V. Niarchos, K. Papadodimas and G. Vos, Large-N correlation functions in N = 2 superconformal QCD, JHEP 01 (2017) 101 [arXiv:1610.07612] [INSPIRE].
  14. [14]
    A. Pini, D. Rodriguez-Gomez and J.G. Russo, Large N correlation functions N = 2 superconformal quivers, JHEP 08 (2017) 066 [arXiv:1701.02315] [INSPIRE].
  15. [15]
    M. Billó, F. Fucito, A. Lerda, J.F. Morales, Ya. S. Stanev and C. Wen, Two-point correlators in N = 2 gauge theories, Nucl. Phys. B 926 (2018) 427 [arXiv:1705.02909] [INSPIRE].
  16. [16]
    M. Billó, F. Galvagno, P. Gregori and A. Lerda, Correlators between Wilson loop and chiral operators in N = 2 conformal gauge theories, JHEP 03 (2018) 193 [arXiv:1802.09813] [INSPIRE].
  17. [17]
    S. Giombi and S. Komatsu, Exact correlators on the Wilson loop in N = 4 SYM: localization, defect CFT and integrability, JHEP 05 (2018) 109 [Erratum ibid. 11 (2018) 123] [arXiv:1802.05201] [INSPIRE].
  18. [18]
    A. Bourget, D. Rodriguez-Gomez and J.G. Russo, A limit for large R-charge correlators in N = 2 theories, JHEP 05 (2018) 074 [arXiv:1803.00580] [INSPIRE].
  19. [19]
    M. Beccaria, On the large R-charge N = 2 chiral correlators and the Toda equation, arXiv:1809.06280 [INSPIRE].
  20. [20]
    S. Hellerman and S. Maeda, On the large R-charge expansion in N = 2 superconformal field theories, JHEP 12 (2017) 135 [arXiv:1710.07336] [INSPIRE].
  21. [21]
    S. Hellerman, S. Maeda, D. Orlando, S. Reffert and M. Watanabe, Universal correlation functions in rank 1 SCFTs, arXiv:1804.01535 [INSPIRE].
  22. [22]
    A. Gerasimov, A. Marshakov, A. Mironov, A. Morozov and A. Orlov, Matrix models of 2D gravity and Toda theory, Nucl. Phys. B 357 (1991) 565 [INSPIRE].
  23. [23]
    I.G. Koh and S. Rajpoot, Finite N = 2 extended supersymmetric field theories, Phys. Lett. B 135 (1984) 397 [INSPIRE].
  24. [24]
    P.S. Howe, K.S. Stelle and P.C. West, A class of finite four-dimensional supersymmetric field theories, Phys. Lett. B 124 (1983) 55 [INSPIRE].
  25. [25]
    B. Fiol, B. Garolera and G. Torrents, Probing N = 2 superconformal field theories with localization, JHEP 01 (2016) 168 [arXiv:1511.00616] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Antoine Bourget
    • 1
  • Diego Rodriguez-Gomez
    • 2
    Email author
  • Jorge G. Russo
    • 3
    • 4
  1. 1.Theoretical Physics, The Blackett LaboratoryImperial College LondonLondonU.K.
  2. 2.Department of PhysicsUniversidad de OviedoOviedoSpain
  3. 3.Institució Catalana de Recerca i Estudis Avançats (ICREA)BarcelonaSpain
  4. 4.Departament de Física Cuántica i Astrofísica and Institut de Ciències del CosmosUniversitat de BarcelonaBarcelonaSpain

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