Skip to main content

GCA in 2d

Abstract

We make a detailed study of the infinite dimensional Galilean Conformal Algebra (GCA) in the case of two spacetime dimensions. Classically, this algebra is precisely obtained from a contraction of the generators of the relativistic conformal symmetry in 2d. Here we find quantum mechanical realisations of the (centrally extended) GCA by considering scaling limits of certain 2d CFTs. These parent CFTs are non-unitary and have their left and right central charges become large in magnitude and opposite in sign. We therefore develop, in parallel to the usual machinery for 2d CFT, many of the tools for the analysis of the quantum mechanical GCA. These include the representation theory based on GCA primaries, Ward identities for their correlation functions and a nonrelativistic Kac table. In particular, the null vectors of the GCA lead to differential equations for the four point function. The solution to these equations in the simplest case is explicitly obtained and checked to be consistent with various requirements.

References

  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. Rodríguez-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]

    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [SPIRES].

    MATH  MathSciNet  ADS  Google Scholar 

  10. [10]

    D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 super Yang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  11. [11]

    S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. [12]

    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 

  13. [13]

    M. Henkel, R. Schott, S. Stoimenov and J. Unterberger, The Poincaré algebra in the context of ageing systems: Lie structure, representations, Appell systems and coherent states, math-ph/0601028 [SPIRES].

  14. [14]

    V.N. Gusyatnikova and V.A. Yumaguzhin, Symmetries and conservation laws of navier-stokes equations, Acta Appl. Math. 15 (1989) 65.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    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 

  16. [16]

    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 

  17. [17]

    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 

  18. [18]

    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 

  19. [19]

    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 

  20. [20]

    M. Alishahiha, A. Davody and A. Vahedi, On AdS/CFT of Galilean conformal field theories, JHEP 08 (2009) 022 [arXiv:0903.3953] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  21. [21]

    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 

  22. [22]

    D. Martelli and Y. Tachikawa, Comments on Galilean conformal field theories and their geometric realization, JHEP 05 (2010) 091 [arXiv:0903.5184] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  23. [23]

    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 

  24. [24]

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

    ADS  Google Scholar 

  25. [25]

    M. Sakaguchi, Super Galilean conformal algebra in AdS/CFT, arXiv:0905.0188 [SPIRES].

  26. [26]

    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 

  27. [27]

    A. Hosseiny and S. Rouhani, Affine extension of Galilean conformal algebra in 2+1 dimensions, arXiv:0909.1203 [SPIRES].

  28. [28]

    K. Hotta, T. Kubota and T. Nishinaka, Galilean conformal algebra in two dimensions and cosmological topologically massive gravity, arXiv:1003.1203 [SPIRES].

  29. [29]

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

    Article  ADS  Google Scholar 

  30. [30]

    S.V. Ketov, Conformal field theory, World Scientific Singapore, Singapore (1995) [SPIRES].

    MATH  Google Scholar 

  31. [31]

    P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York, U.S.A. (1997) [SPIRES].

    MATH  Google Scholar 

  32. [32]

    I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, 7th edition, Academic Press, London U.K. (2007).

    Google Scholar 

  33. [33]

    S. Moriguchi, K. Udagawa and S. Hitotsumatsu, Iwanami Sugaku Koshiki III, 17th edition, Iwanami Shoten, Japan (2001).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Arjun Bagchi.

Additional information

ArXiv ePrint: 0912.1090

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), 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

Bagchi, A., Gopakumar, R., Mandal, I. et al. GCA in 2d. J. High Energ. Phys. 2010, 4 (2010). https://doi.org/10.1007/JHEP08(2010)004

Download citation

Keywords

  • Field Theories in Lower Dimensions
  • Conformal and W Symmetry