Advertisement

Exactly marginal deformations and global symmetries

  • Daniel Green
  • Zohar Komargodski
  • Nathan Seiberg
  • Yuji TachikawaEmail author
  • Brian Wecht
Article

Abstract

We study the problem of finding exactly marginal deformations of \( \mathcal{N} = 1 \) superconformal field theories in four dimensions. We find that the only way a marginal chiral operator can become not exactly marginal is for it to combine with a conserved current multiplet. Additionally, we find that the space of exactly marginal deformations, also called the “conformal manifold,” is the quotient of the space of marginal couplings by the complexified continuous global symmetry group. This fact explains why exactly marginal deformations are ubiquitous in \( \mathcal{N} = 1 \) theories. Our method turns the problem of enumerating exactly marginal operators into a problem in group theory, and substantially extends and simplifies the previous analysis by Leigh and Strassler. We also briefly discuss how to apply our analysis to \( \mathcal{N} = 2 \) theories in three dimensions.

Keywords

Supersymmetric gauge theory Renormalization Group Supersymmetry and Duality Global Symmetries 

References

  1. [1]
    A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [Pisma Zh. Eksp. Teor. Fiz. 43 (1986) 565] [SPIRES].MathSciNetADSGoogle Scholar
  2. [2]
    L.J. Dixon, Some World Sheet Properties of Superstring Compactifications, on Orbifolds and Otherwise, Lectures given at the ICTP Summer Workshop in High Energy Phsyics and Cosmology, Trieste, Italy, Jun 29 – Aug 7 (1987), in Superstrings, unified theories and cosmology, World Scientific, Singapore (1988).Google Scholar
  3. [3]
    N. Seiberg, Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys. B 303 (1988) 286 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    D. Kutasov, Geometry on the space of conformal field theories and contact terms, Phys. Lett. B 220 (1989) 153 [SPIRES].MathSciNetADSGoogle Scholar
  5. [5]
    S. Cecotti, S. Ferrara and L. Girardello, A topological formula for the Kähler potential of 4D N = 1, N = 2 strings and its implications for the moduli problem, Phys. Lett. B 213 (1988) 443 [SPIRES].MathSciNetADSGoogle Scholar
  6. [6]
    S. Cecotti, S. Ferrara and L. Girardello, Geometry of Type II Superstrings and the Moduli of Superconformal Field Theories, Int. J. Mod. Phys. A 4 (1989) 2475 [SPIRES].MathSciNetADSGoogle Scholar
  7. [7]
    L.J. Dixon, V. Kaplunovsky and J. Louis, On Effective Field Theories Describing (2,2) Vacua of the Heterotic String, Nucl. Phys. B 329 (1990) 27 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    A. Strominger, Special geometry, Commun. Math. Phys. 133 (1990) 163 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  9. [9]
    R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 95 [hep-th/9503121] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  10. [10]
    O. Aharony, B. Kol and S. Yankielowicz, On exactly marginal deformations of N = 4 SYM and type IIB supergravity on AdS 5 × S 5, JHEP 06 (2002) 039 [hep-th/0205090] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  11. [11]
    B. Kol, On conformal deformations, JHEP 09 (2002) 046 [hep-th/0205141] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  12. [12]
    Y. Tachikawa, Five-dimensional supergravity dual of a-maximization, Nucl. Phys. B 733 (2006) 188 [hep-th/0507057] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    E. Barnes, E. Gorbatov, K.A. Intriligator and J. Wright, Current correlators and AdS/CFT geometry, Nucl. Phys. B 732 (2006) 89 [hep-th/0507146] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    V. Asnin, On metric geometry of conformal moduli spaces of four- dimensional superconformal theories, arXiv:0912.2529 [SPIRES].
  15. [15]
    B. Kol, unpublished notes.Google Scholar
  16. [16]
    P.C. Argyres, K.A. Intriligator, R.G. Leigh and M.J. Strassler, On inherited duality in N = 1 D = 4 supersymmetric gauge theories, JHEP 04 (2000) 029 [hep-th/9910250] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  17. [17]
    A.E. Nelson and M.J. Strassler, Exact results for supersymmetric renormalization and the supersymmetric flavor problem, JHEP 07 (2002) 021 [hep-ph/0104051] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  18. [18]
    V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Exact Gell-Mann-Low Function of Supersymmetric Yang-Mills Theories from Instanton Calculus, Nucl. Phys. B 229 (1983) 381 [SPIRES].CrossRefADSGoogle Scholar
  19. [19]
    N. Seiberg, Naturalness Versus Supersymmetric Non-renormalization Theorems, Phys. Lett. B 318 (1993) 469 [hep-ph/9309335] [SPIRES].MathSciNetADSGoogle Scholar
  20. [20]
    P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  21. [21]
    P. Meade, N. Seiberg and D. Shih, General Gauge Mediation, Prog. Theor. Phys. Suppl. 177 (2009) 143 [arXiv:0801.3278] [SPIRES].zbMATHCrossRefADSGoogle Scholar
  22. [22]
    K.A. Intriligator and N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. Proc. Suppl. 45BC (1996) 1 [hep-th/9509066] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  23. [23]
    C. Lucchesi and G. Zoupanos, All-order Finiteness in N = 1 SYM Theories: Criteria and Applications, Fortschr. Phys. 45 (1997) 129 [hep-ph/9604216] [SPIRES].zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    D. Gaiotto and X. Yin, Notes on superconformal Chern-Simons-matter theories, JHEP 08 (2007) 056 [arXiv:0704.3740] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  25. [25]
    N. Akerblom, C. Sämann and M. Wolf, Marginal Deformations and 3-Algebra Structures, Nucl. Phys. B 826 (2010) 456 [arXiv:0906.1705] [SPIRES].CrossRefADSGoogle Scholar
  26. [26]
    M.S. Bianchi, S. Penati and M. Siani, Infrared stability of ABJ-like theories, JHEP 01 (2010) 080 [arXiv:0910.5200] [SPIRES].CrossRefGoogle Scholar
  27. [27]
    M.S. Bianchi, S. Penati and M. Siani, Infrared Stability of N = 2 Chern-Simons Matter Theories, JHEP 05 (2010) 106 [arXiv:0912.4282] [SPIRES].CrossRefGoogle Scholar
  28. [28]
    C.-M. Chang and X. Yin, Families of Conformal Fixed Points of N = 2 Chern-Simons- Matter Theories, JHEP 05 (2010) 108 [1002.0568] [SPIRES].CrossRefGoogle Scholar
  29. [29]
    M.J. Strassler, On renormalization group flows and exactly marginal operators in three dimensions, hep-th/9810223 [SPIRES].
  30. [30]
    S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [SPIRES].Google Scholar
  31. [31]
    N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  32. [32]
    I.R. Klebanov and E. Witten, Superconformal field theory on threebranes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  33. [33]
    S. Benvenuti and A. Hanany, Conformal manifolds for the conifold and other toric field theories, JHEP 08 (2005) 024 [hep-th/0502043] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  34. [34]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  35. [35]
    D. Kutasov, New results on the ’a-theorem’ in four dimensional supersymmetric field theory, hep-th/0312098 [SPIRES].
  36. [36]
    E. Barnes, K.A. Intriligator, B. Wecht and J. Wright, Evidence for the strongest version of the 4D a-theorem, via a-maximization along RG flows, Nucl. Phys. B 702 (2004) 131 [hep-th/0408156] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  37. [37]
    D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c-theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [SPIRES].zbMATHMathSciNetGoogle Scholar
  38. [38]
    K. Papadodimas, Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories, arXiv:0910.4963 [SPIRES].
  39. [39]
    M. Flato and C. Fronsdal, Representations of conformal supersymmetry, Lett. Math. Phys. 8 (1984) 159 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  40. [40]
    V.K. Dobrev and V.B. Petkova, All Positive Energy Unitary Irreducible Representations of Extended Conformal Supersymmetry, Phys. Lett. B 162 (1985) 127 [SPIRES].MathSciNetADSGoogle Scholar
  41. [41]
    H. Osborn, N = 1 superconformal symmetry in four-dimensional quantum field theory, Annals Phys. 272 (1999) 243 [hep-th/9808041] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  • Daniel Green
    • 1
  • Zohar Komargodski
    • 1
  • Nathan Seiberg
    • 1
  • Yuji Tachikawa
    • 1
    Email author
  • Brian Wecht
    • 1
  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.

Personalised recommendations