Skip to main content

Supersymmetric extension of GCA in 2d


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.


  1. [1]

    C.R. Hagen, Scale and conformal transformations in galilean-covariant field theory, Phys. Rev. D5 (1972) 377 [SPIRES].

    ADS  Google Scholar 

  2. [2]

    U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta 45 (1972) 802 [SPIRES].

    MathSciNet  Google Scholar 

  3. [3]

    M. Henkel, Schrödinger invariance in strongly anisotropic critical systems, J. Stat. Phys. 75 (1994) 1023 [hep-th/9310081] [SPIRES].

    MATH  Article  ADS  Google Scholar 

  4. [4]

    Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  5. [5]

    J . Negro, M.A. del Olmo and A. Rodriguez-Marco, Non-relativistic conformal groups I, J. Math. Phys. 38 (1997) 3786.

    MATH  Article  MathSciNet  ADS  Google 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].

    MathSciNet  ADS  Google 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].

    Article  MathSciNet  ADS  Google Scholar 

  8. [8]

    A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [SPIRES].

    Article  MathSciNet  ADS  Google 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].

    Article  MathSciNet  Google 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].

    Article  MathSciNet  ADS  Google Scholar 

  12. [12]

    I. Fouxon and Y. Oz, CFT hydrodynamics: Symmetries, exact solutions and gravity, JHEP 03 (2009) 120 [arXiv:0812.1266] [SPIRES].

    Article  MathSciNet  ADS  Google 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].

    Article  MathSciNet  ADS  Google 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].

    ADS  Google 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].

    MathSciNet  ADS  Google 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].

    MathSciNet  ADS  Google 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].

    Article  ADS  Google Scholar 

  19. [19]

    A. Bagchi and I. Mandal, Supersymmetric extension of Galilean conformal algebras, Phys. Rev. D 80 (2009) 086011 [arXiv:0905.0580] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  20. [20]

    J .A. de Azcarraga and J . Lukierski, Galilean superconformal symmetries, Phys. Lett. B 678 (2009) 411 [arXiv:0905.0141] [SPIRES].

    ADS  Google Scholar 

  21. [21]

    M. Sakaguchi, Super Galilean conformal algebra in AdS/CFT, J. Math. Phys. 51 (2010) 042301 [arXiv:0905.0188] [SPIRES].

    Article  MathSciNet  ADS  Google 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].

    Article  ADS  Google 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].

    Article  MathSciNet  ADS  Google Scholar 

  24. [24]

    A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP 08 (2010) 004 [arXiv:0912.1090] [SPIRES].

    Article  ADS  Google 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].

    Article  MathSciNet  ADS  Google 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].

    MathSciNet  ADS  Google 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].

    MathSciNet  ADS  Google Scholar 

  28. [28]

    Z.A. Qiu, Supersymmetry, two-dimensional critical phenomena and the tricritical Ising model, Nucl. Phys. B 270 (1986) 205 [SPIRES].

    Article  ADS  Google 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].

    MathSciNet  ADS  Google Scholar 

  30. [30]

    V.G. Kac, Highest weight representations of infinite-dimensional Lie algebras, proceedings of the International Congress of Mathematicians, Helsinki Finland (1978).

  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].

    MATH  Article  MathSciNet  ADS  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Ipsita Mandal.

Additional information

ArXiv ePrint: hep-th/1003.0209

Rights and permissions

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

Reprints and Permissions

About this article

Cite this article

Mandal, I. Supersymmetric extension of GCA in 2d. J. High Energ. Phys. 2010, 18 (2010).

Download citation


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