Journal of High Energy Physics

, 2010:18 | Cite as

Supersymmetric extension of GCA in 2d

Open Access


We derive the infinite dimensional Supersymmetric Galilean Conformal Algebra (SGCA) in the case of two spacetime dimensions by performing group contraction on 2d superconformal algebra. We also obtain the representations of the generators in terms of superspace coordinates. Here we find realisations of the SGCA by considering scaling limits of certain 2d SCFTs which are non-unitary and have their left and right central charges become large in magnitude and opposite in sign. We focus on the Neveu-Schwarz sector of the parent SCFTs and develop, in parallel to the GCA studies recently in (hepth/0912.1090), the representation theory based on SGCA primaries, Ward identities for their correlation functions and their descendants which are null states.


Field Theories in Lower Dimensions Conformal and W Symmetry AdS-CFT Correspondence 


  1. [1]
    C.R. Hagen, Scale and conformal transformations in galilean-covariant field theory, Phys. Rev. D5 (1972) 377 [SPIRES].ADSGoogle Scholar
  2. [2]
    U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972) 802 [SPIRES].MathSciNetGoogle Scholar
  3. [3]
    M. Henkel, Schrödinger invariance in strongly anisotropic critical systems, J. Stat. Phys. 75 (1994) 1023 [hep-th/9310081] [SPIRES]. MATHCrossRefADSGoogle Scholar
  4. [4]
    Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [SPIRES].MathSciNetADSGoogle Scholar
  5. [5]
    J . Negro, M.A. del Olmo and A. Rodriguez-Marco, Non-relativistic conformal groups I, J. Math. Phys. 38 (1997) 3786. MATHCrossRefMathSciNetADSGoogle Scholar
  6. [6]
    J. Lukierski, P.C. Stichel and W.J. Zakrzewski, Exotic Galilean conformal symmetry and its dynamical realisations, Phys. Lett. A 357 (2006) 1 [hep-th/0511259] [SPIRES].MathSciNetADSGoogle Scholar
  7. [7]
    J . Gomis, J . Gomis and K. Kamimura, Non-relativistic superstrings: A new soluble sector of AdS 5 ×S 5, JHEP 12 (2005) 024 [hep-th/0507036] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  8. [8]
    A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  9. [9]
    M. Henkel, Phenomenology of local scale invariance: From conformal invariance to dynamical scaling, Nucl. Phys. B 641 (2002) 405 [hep-th/0205256] [SPIRES].CrossRefMathSciNetGoogle Scholar
  10. [10]
    M. Henkel, R. Schott, S. Stoimenov and J. Unterberger, T he Poincaré algebra in the context of ageing systems: Lie structure, representations, appell systems and coherent states, math-ph/0601028 [SPIRES].
  11. [11]
    I. Fouxon and Y. Oz, Conformal field theory as microscopic dynamics of incompressible Euler and Navier-Stokes equations, Phys. Rev. Lett. 101 (2008) 261602 [arXiv:0809.4512] [SPIRES]. CrossRefMathSciNetADSGoogle Scholar
  12. [12]
    I. Fouxon and Y. Oz, CFT hydrodynamics: Symmetries, exact solutions and gravity, JHEP 03 (2009) 120 [arXiv:0812.1266] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    S. Bhattacharyya, S. Minwalla and S.R. Wadia, The incompressible non-relativistic Navier-Stokes equation from gravity, JHEP 08 (2009) 059 [arXiv:0810.1545] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    P.A. Horvathy and P.M. Zhang, Non-relativistic conformal symmetries in fluid mechanics, Eur. Phys. J. C 65 (2010) 607 [arXiv:0906.3594] [SPIRES].ADSGoogle Scholar
  15. [15]
    C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys. A 42 (2009) 465206 [arXiv:0904.0531] [SPIRES].MathSciNetADSGoogle Scholar
  16. [16]
    M. Alishahiha, A. Davody and A. Vahedi, On AdS/CFT of Galilean conformal field theories, [arXiv:0903.3953] [SPIRES].
  17. [17]
    A. Bagchi and I. Mandal, On representations and correlation functions of Galilean conformal algebras, Phys. Lett. B 675 (2009) 393 [arXiv:0903.4524] [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    D. Martelli and Y. Tachikawa, Comments on Galilean conformal field theories and their geometric realization, JHEP 05 (2010) 091 [arXiv:0903.5184] [SPIRES].CrossRefADSGoogle Scholar
  19. [19]
    A. Bagchi and I. Mandal, Supersymmetric extension of Galilean conformal algebras, Phys. Rev. D 80 (2009) 086011 [arXiv:0905.0580] [SPIRES].MathSciNetADSGoogle Scholar
  20. [20]
    J .A. de Azcarraga and J . Lukierski, Galilean superconformal symmetries, Phys. Lett. B 678 (2009) 411 [arXiv:0905.0141] [SPIRES].ADSGoogle Scholar
  21. [21]
    M. Sakaguchi, Super Galilean conformal algebra in AdS/CFT, J. Math. Phys. 51 (2010) 042301 [arXiv:0905.0188] [SPIRES]. CrossRefMathSciNetADSGoogle Scholar
  22. [22]
    A. Mukhopadhyay, A covariant form of the Navier-Stokes equation for the Galilean conformal algebra, JHEP 01 (2010) 100 [arXiv:0908.0797] [SPIRES].CrossRefADSGoogle Scholar
  23. [23]
    A. Hosseiny and S. Rouhani, Affine extension of Galilean conformal algebra in 2+1 dimensions, J. Math. Phys. 51 (2010) 052307 [arXiv:0909.1203] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  24. [24]
    A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP 08 (2010) 004 [arXiv:0912.1090] [SPIRES].CrossRefADSGoogle Scholar
  25. [25]
    D. Friedan, Z.-a. Qiu and S.H. Shenker, Conformal invariance, unitarity and two-dimensional critical exponents, Phys. Rev. Lett. 52 (1984) 1575 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  26. [26]
    M.A. Bershadsky, V.G. Knizhnik and M.G. Teitelman, Superconformal symmetry in two-dimensions, Phys. Lett. B 151 (1985) 31 [SPIRES].MathSciNetADSGoogle Scholar
  27. [27]
    D. Friedan, Z. Qiu and S.H. Shenker, Superconformal invariance in two dimensions and the tricritical ising model, Phys. Lett. B 151 (1985) 37 [SPIRES].MathSciNetADSGoogle Scholar
  28. [28]
    Z.A. Qiu, Supersymmetry, two-dimensional critical phenomena and the tricritical Ising model, Nucl. Phys. B 270 (1986) 205 [SPIRES].CrossRefADSGoogle Scholar
  29. [29]
    G.M. Sotkov and M.S. Stanishkov, N=1 superconformal operator product expansions and superfield fusion rules, Phys. Lett. B 177 (1986) 361 [SPIRES].MathSciNetADSGoogle Scholar
  30. [30]
    V.G. Kac, Highest weight representations of infinite-dimensional Lie algebras, proceedings of the International Congress of Mathematicians, Helsinki Finland (1978).Google Scholar
  31. [31]
    P. Goddard, A. Kent and D.I. Olive, Unitary representations of the Virasoro and supervirasoro algebras, Commun. Math. Phys. 103 (1986) 105 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar

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© The Author(s) 2010

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Harish-Chandra Research InstituteJhusiIndia
  2. 2.LPTHEUniversite Pierre et Marie CurieParis Cedex 05France

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