Journal of High Energy Physics

, 2014:61 | Cite as

Galilean conformal electrodynamics

  • Arjun BagchiEmail author
  • Rudranil Basu
  • Aditya Mehra
Open Access
Regular Article - Theoretical Physics


Maxwell’s Electrodynamics admits two distinct Galilean limits called the Electric and Magnetic limits. We show that the equations of motion in both these limits are invariant under the Galilean Conformal Algebra in D = 4, thereby exhibiting non-relativistic conformal symmetries. Remarkably, the symmetries are infinite dimensional and thus Galilean Electrodynamics give us the first example of an infinitely extended Galilean Conformal Field Theory in D > 2. We examine details of the theory by looking at purely non-relativistic conformal methods and also use input from the limit of the relativistic theory.


Conformal and W Symmetry Gauge-gravity correspondence AdS-CFT Correspondence 


Open Access

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  1. [1]
    G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
  2. [2]
    L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    S. Ferrara, A.F. Grillo, G. Parisi and R. Gatto, Covariant expansion of the conformal four-point function, Nucl. Phys. B 49 (1972) 77 [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    R. Rattazzi, S. Rychkov and A. Vichi, Bounds in 4D conformal field theories with global symmetry, J. Phys. A 44 (2011) 035402 [arXiv:1009.5985] [INSPIRE].ADSMathSciNetGoogle Scholar
  11. [11]
    D. Poland, D. Simmons-Duffin and A. Vichi, Carving out the space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].ADSGoogle Scholar
  13. [13]
    A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP 08 (2010) 004 [arXiv:0912.1090] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    M. Le Bellac and J.-M. Lévy-Leblond, Galilean electromagnetism, Nuovo Cim. 14B (1973) 217.ADSCrossRefGoogle Scholar
  16. [16]
    R. Jackiw and S.-Y. Pi, Tutorial on scale and conformal symmetries in diverse dimensions, J. Phys. A 44 (2011) 223001 [arXiv:1101.4886] [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    S. El-Showk, Y. Nakayama and S. Rychkov, What Maxwell theory in D ≠ 4 teaches us about scale and conformal invariance, Nucl. Phys. B 848 (2011) 578 [arXiv:1101.5385] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    A. Bagchi and I. Mandal, On representations and correlation functions of Galilean conformal algebras, Phys. Lett. B 675 (2009) 393 [arXiv:0903.4524] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    A. Bagchi, Tensionless strings and Galilean conformal algebra, JHEP 05 (2013) 141 [arXiv:1303.0291] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    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] [INSPIRE].
  21. [21]
    J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B 303 (1988) 226 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    C. Bunster and M. Henneaux, Duality invariance implies Poincaré invariance, Phys. Rev. Lett. 110 (2013) 011603 [arXiv:1208.6302] [INSPIRE].ADSCrossRefGoogle Scholar
  23. [23]
    C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Class. Quant. Grav. 31 (2014) 085016 [arXiv:1402.0657] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  26. [26]
    G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [INSPIRE].CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    A. Bagchi, The BMS/GCA correspondence, arXiv:1006.3354 [INSPIRE].
  28. [28]
    A. Bagchi and R. Fareghbal, BMS/GCA redux: towards flatspace holography from non-relativistic symmetries, JHEP 10 (2012) 092 [arXiv:1203.5795] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    G. Barnich, A. Gomberoff and H.A. Gonzalez, The flat limit of three dimensional asymptotically anti-de Sitter spacetimes, Phys. Rev. D 86 (2012) 024020 [arXiv:1204.3288] [INSPIRE].ADSGoogle Scholar
  30. [30]
    A. Bagchi, S. Detournay and D. Grumiller, Flat-space chiral gravity, Phys. Rev. Lett. 109 (2012) 151301 [arXiv:1208.1658] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    A. Bagchi, S. Detournay, R. Fareghbal and J. Simon, Holography of 3d flat cosmological horizons, Phys. Rev. Lett. 110 (2013) 141302 [arXiv:1208.4372] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10 (2012) 095 [arXiv:1208.4371] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    G. Barnich, A. Gomberoff and H.A. González, Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory, Phys. Rev. D 87 (2013) 124032 [arXiv:1210.0731] [INSPIRE].ADSGoogle Scholar
  34. [34]
    A. Bagchi, S. Detournay, D. Grumiller and J. Simon, Cosmic evolution from phase transition of three-dimensional flat space, Phys. Rev. Lett. 111 (2013) 181301 [arXiv:1305.2919] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Spin-3 gravity in three-dimensional flat space, Phys. Rev. Lett. 111 (2013) 121603 [arXiv:1307.4768] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    H.A. Gonzalez, J. Matulich, M. Pino and R. Troncoso, Asymptotically flat spacetimes in three-dimensional higher spin gravity, JHEP 09 (2013) 016 [arXiv:1307.5651] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  37. [37]
    A. Bagchi and R. Basu, 3D flat holography: entropy and logarithmic corrections, JHEP 03 (2014) 020 [arXiv:1312.5748] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    R. Fareghbal and A. Naseh, Flat-space energy-momentum tensor from BMS/GCA correspondence, JHEP 03 (2014) 005 [arXiv:1312.2109] [INSPIRE].ADSCrossRefGoogle Scholar
  39. [39]
    C. Krishnan, A. Raju and S. Roy, A Grassmann path from AdS 3 to flat space, JHEP 03 (2014) 036 [arXiv:1312.2941] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    S. Detournay, D. Grumiller, F. Scholler and J. Simon, Variational principle and 1-point functions in 3-dimensional flat space Einstein gravity, Phys. Rev. D 89 (2014) 084061 [arXiv:1402.3687] [INSPIRE].ADSGoogle Scholar
  41. [41]
    C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [arXiv:1402.5894] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups, J. Phys. A 47 (2014) 335204 [arXiv:1403.4213] [INSPIRE].MathSciNetGoogle Scholar
  43. [43]
    C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys. A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].ADSMathSciNetGoogle Scholar
  44. [44]
    A. Bagchi and I. Mandal, Supersymmetric extension of Galilean conformal algebras, Phys. Rev. D 80 (2009) 086011 [arXiv:0905.0580] [INSPIRE].ADSMathSciNetGoogle Scholar
  45. [45]
    M. Sakaguchi, Super Galilean conformal algebra in AdS/CFT, J. Math. Phys. 51 (2010) 042301 [arXiv:0905.0188] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  46. [46]
    J.A. de Azcarraga and J. Lukierski, Galilean superconformal symmetries, Phys. Lett. B 678 (2009) 411 [arXiv:0905.0141] [INSPIRE].ADSCrossRefGoogle Scholar
  47. [47]
    D. Grumiller, M. Leston and D. Vassilevich, Anti-de Sitter holography for gravity and higher spin theories in two dimensions, Phys. Rev. D 89 (2014) 044001 [arXiv:1311.7413] [INSPIRE].ADSGoogle Scholar
  48. [48]
    C.R. Hagen, Scale and conformal transformations in Galilean-covariant field theory, Phys. Rev. D 5 (1972) 377 [INSPIRE].ADSGoogle Scholar
  49. [49]
    U. Niederer, The maximal kinematical invariance group of the free Schrödinger equation., Helv. Phys. Acta 45 (1972) 802 [INSPIRE].MathSciNetGoogle Scholar
  50. [50]
    Y. Nishida and D.T. Son, Nonrelativistic conformal field theories, Phys. Rev. D 76 (2007) 086004 [arXiv:0706.3746] [INSPIRE].ADSMathSciNetGoogle Scholar
  51. [51]
    D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].ADSMathSciNetGoogle Scholar
  52. [52]
    K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  53. [53]
    M. Henkel, Schrödinger invariance in strongly anisotropic critical systems, J. Statist. Phys. 75 (1994) 1023 [hep-th/9310081] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  54. [54]
    D. Martelli and Y. Tachikawa, Comments on Galilean conformal field theories and their geometric realization, JHEP 05 (2010) 091 [arXiv:0903.5184] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  55. [55]
    D.T. Son, Newton-Cartan geometry and the quantum Hall effect, arXiv:1306.0638 [INSPIRE].
  56. [56]
    R. Andringa, E.A. Bergshoeff, J. Rosseel and E. Sezgin, 3D Newton-Cartan supergravity, Class. Quant. Grav. 30 (2013) 205005 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  57. [57]
    V. de Alfaro, S. Fubini and G. Furlan, Conformal invariance in quantum mechanics, Nuovo Cim. A 34 (1976) 569 [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    S. Fedoruk, E. Ivanov and J. Lukierski, Galilean conformal mechanics from nonlinear realizations, Phys. Rev. D 83 (2011) 085013 [arXiv:1101.1658] [INSPIRE].ADSGoogle Scholar
  59. [59]
    C. Chamon, R. Jackiw, S.-Y. Pi and L. Santos, Conformal quantum mechanics as the CFT 1 dual to AdS 2, Phys. Lett. B 701 (2011) 503 [arXiv:1106.0726] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  60. [60]
    A. Strominger, Asymptotic symmetries of Yang-Mills theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Indian Institute of Science Education and ResearchPuneIndia
  2. 2.Institute of Theoretical PhysicsVienna University of TechnologyViennaAustria
  3. 3.The Institute of Mathematical SciencesChennaiIndia

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