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On exact correlation functions in SU(N) \( \mathcal{N}=2 \) superconformal QCD

A preprint version of the article is available at arXiv.

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.

References

  1. [1]

    S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  2. [2]

    K. Papadodimas, Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  3. [3]

    W. Lerche, C. Vafa and N.P. Warner, Chiral Rings in N = 2 Superconformal Theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  4. [4]

    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].

    Article  ADS  MathSciNet  Google Scholar 

  5. [5]

    S. Cecotti and C. Vafa, Exact results for supersymmetric σ-models, Phys. Rev. Lett. 68 (1992) 903 [hep-th/9111016] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. [6]

    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].

    Article  ADS  Google Scholar 

  7. [7]

    E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere Partition Functions and the Zamolodchikov Metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  8. [8]

    J. Gomis and N. Ishtiaque, Kähler potential and ambiguities in 4d \( \mathcal{N}=2 \) SCFTs, JHEP 04 (2015) 169 [arXiv:1409.5325] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  9. [9]

    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. [10]

    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].

    Article  ADS  Google Scholar 

  11. [11]

    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].

    MathSciNet  MATH  Google Scholar 

  12. [12]

    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].

    ADS  MathSciNet  Google Scholar 

  13. [13]

    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].

  14. [14]

    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].

    Article  ADS  MathSciNet  Google Scholar 

  15. [15]

    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].

    Article  ADS  MathSciNet  Google Scholar 

  16. [16]

    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].

    Article  ADS  MathSciNet  Google Scholar 

  17. [17]

    A. Petkou and K. Skenderis, A nonrenormalization theorem for conformal anomalies, Nucl. Phys. B 561 (1999) 100 [hep-th/9906030] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  18. [18]

    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].

    Article  ADS  MathSciNet  Google Scholar 

  19. [19]

    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].

    Article  ADS  MathSciNet  Google Scholar 

  20. [20]

    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].

    Article  ADS  Google Scholar 

  21. [21]

    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].

    Article  ADS  Google Scholar 

  22. [22]

    D. Binosi and L. Theussl, JaxoDraw: A graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun. 161 (2004) 76 [hep-ph/0309015] [INSPIRE].

    Article  ADS  Google Scholar 

  23. [23]

    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].

    Article  ADS  Google Scholar 

  24. [24]

    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].

    Article  ADS  MathSciNet  Google Scholar 

  25. [25]

    R. Andree and D. Young, Wilson Loops in N = 2 Superconformal Yang-Mills Theory, JHEP 09 (2010) 095 [arXiv:1007.4923] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

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Correspondence to Vasilis Niarchos.

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ArXiv ePrint: 1508.03077

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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

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Keywords

  • Supersymmetric gauge theory
  • Extended Supersymmetry
  • Gauge Symmetry
  • Nonperturbative Effects