Journal of High Energy Physics

, 2019:41 | Cite as

Uniqueness of Galilean conformal electrodynamics and its dynamical structure

  • Kinjal Banerjee
  • Rudranil Basu
  • Akhila MohanEmail author
Open Access
Regular Article - Theoretical Physics


We investigate the existence of action for both the electric and magnetic sectors of Galilean Electrodynamics using Helmholtz conditions. We prove the existence of unique action in magnetic limit with the addition of a scalar field in the system. The check also implies the non existence of action in the electric sector of Galilean electrodynamics. Dirac constraint analysis of the theory reveals that there are no local degrees of freedom in the system. Further, the theory enjoys a reduced but an infinite dimensional subalgebra of Galilean conformal symmetry algebra as global symmetries. The full Galilean conformal algebra however is realized as canonical symmetries on the phase space. The corresponding algebra of Hamilton functions acquire a state dependent central charge.


Conformal Field Theory Space-Time Symmetries Conformal and W Sym­metry Global Symmetries 


Open Access

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  1. [1]
    N. Beisert et al., Review of AdSjCFT integrability: an overmew, Lett. Math. Phys.99 (2012) 3 [arXiv: 1012 .3982] [INSPIRE].
  2. [2]
    N. Beisert, A. Garus and M. Rosso, Yangian symmetry for the action of planar N = 4 super Yang-Mills and N = 6 super Chern-Simons theories, Phys. Rev.D 98 (2018) 046006 [arXiv:1803.06310] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    H.P. Kuenzle, Galilei and Lorentz structures on space-time - comparison of the corresponding geometry and physics, Ann. Inst. H. Poincare Phys. Theor.17 (1972) 337.MathSciNetGoogle Scholar
  4. [4]
    A. Bagchi and R. Gopakumar, Galilean conformal algebras and AdS/CFT, JHEP07 (2009) 037 [arXiv:0902.1385] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    N. Brambilla, D. Gromes and A. Vairo, Poincare invariance constraints on NRQCD and potential NRQCD, Phys. Lett.B 576 (2003) 314 [hep-ph/0306107] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    M. Geracie, D.T. Son, C. Wu and S.-F. Wu, Spacetime symmetries of the quantum Hall effect, Phys. Rev.D 91 (2015) 045030 [arXiv:1407.1252] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    A. Mohan, K. Madhu and V. Sunilkumar, Lifshitz-type gauge theory with N = 2 supersymmetry, Int. J. Mod. Phys.A 34 (2019) 1950080 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    S. Chapman, Y. Oz and A. Raviv-Moshe, On supersymmetric Lifshit z field theories, JHEP10 (2015) 162 [arXiv:1508.03338] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    M. Gomes, J. Queiruga and A.J. da Silva, Lorentz breaking supersymmetry and Hofava-Lifshit z-like models, Phys. Rev.D 92 (2015) 025050 [arXiv:1506.01331] [INSPIRE].ADSGoogle Scholar
  10. [10]
    M. Le Bellac and J.M. Levy-Leblond, Galilean el ectromagnetism, Nuovo Cim.14 (1973) 217.Google Scholar
  11. [11]
    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].ADSMathSciNetCrossRefGoogle Scholar
  12. [12]
    C. Duval and P.A. Horvathy, Non-relativistic conformal symmetries and Newton-Cartan structures, J. Phys.A 42 (2009) 465206 [arXiv:0904.0531] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  13. [13]
    A. Bagchi, R. Basu and A. Mehra, Galilean conformal electrodynamics, JHEP11 (2014) 061 [arXiv:1408.0810] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Galilean Yang-Mills theory, JHEP04 (2016) 051 [arXiv:1512.08375] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  15. [15]
    A. Bagchi, J. Chakrabortty and A. Mehra, Galilean field theories and conformal structure, JHEP04 (2018) 144 [arXiv:1712.05631] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    E. Bergshoeff, J. Rosseel and T. Zojer, Non-relativistic fields from arbitrary contracting backgrounds, Class. Quant. Grav.33 (2016) 175010 [arXiv:1512.06064] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    G. Festuccia, D. Hansen, J. Hartong and N.A. Obers, Symmetries and couplings of non-relativistic electrodynamics, JHEP11 (2016) 037 [arXiv:1607.01753] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  18. [18]
    C. Batlle, J. Gomis and D. Not, Extended Galilean symmetries of non-relativistic strings, JHEP02 (2017) 049 [arXiv:1611.00026] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    J. Kluson, Remark about non-relativistic string in Newton-Cartan background and null reduction, JHEP05 (2018) 041 [arXiv:1803.07336] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    J. Kluson, Canonical formalism of nonrelativistic theories coupled to Newton-Cartan gravity, Phys. Rev.D 98 (2018) 066014 [arXiv:1805.12392] [INSPIRE].ADSMathSciNetGoogle Scholar
  21. [21]
    G. Festuccia, D. Hansen, J. Hartong and N.A. Obers, Torsional Newton-Cartan geometry from the Noether procedure, Phys. Rev.D 94 (2016) 105023 [arXiv:1607.01926] [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    E.A. Bergshoeff and J. Rosseel, A new look at Newton-Cartan gravity, Int. J. Mod. Phys.A 31 (2016) 1630040 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    G. Morandi, C. Ferrario, G. LoVecchio, G. Marmo and C. Rubano, The inverse problem in the calculus of variations and the geometry of the tangent bundle, Phys. Rept.188 (1990) 147 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    D.R. Davis, The inverse problem of the calculus of variations in higher space, Trans. Amer. Math. Soc.30 (1928) 710.MathSciNetCrossRefGoogle Scholar
  25. [25]
    N. Kushagra and K. Banerjee, A brief review of Helmholt z conditions, arXiv:1602.01563.
  26. [26]
    M. Henneaux, On the inverse problem of the calculus of variations in field theory, J. Phys.A 17 (1984) 75.ADSMathSciNetzbMATHGoogle Scholar
  27. [27]
    D.R. Davis, The inverse problem of the calculus of variations in a space of (n + 1) dimensions, Bull. Amer. Math. Soc.35 (1929) 371.MathSciNetCrossRefGoogle Scholar
  28. [28]
    J. Douglas, Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc.50 (1941) 71.MathSciNetCrossRefGoogle Scholar
  29. [29]
    C. Crnkovic and E. Witten, Covariant description of canonical formalism in geometrical theories, in Three hundred years of gravitation, S.W. Hawking and W. Israel eds., (1987), pg. 676 [INSPIRE].
  30. [30]
    A. Bagchi, A. Banerjee and P. Parekh, Tensionless path from closed to open strings, Phys. Rev. Lett.123 (2019) 111601 [arXiv:1905.11732] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups, J. Phys.A 47 (2014) 335204 [arXiv:1403.4213] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  32. [32]
    R. Basu and U.N. Chowdhury, Dynamical structure of Carrollian electrodynamics, JHEP04 (2018) 111 [arXiv:1802.09366] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys.104 (1986) 207 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    S. Deser, R. Jackiw and S. Templeton, Three-dimensional massive gauge theories, Phys. Rev. Lett.48 (1982) 975 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    K. Jensen, Anomalies for Galilean fields, SciPost Phys.5 (2018) 005 [arXiv:1412.7750] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A. Jain, Galilean anomalies and their effect on hydrodynamics, Phys. Rev.D 93 (2016) 065007 [arXiv:1509.05777] [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    A. Bagchi, A. Mehra and P. Nandi, Field theories with conformal Carrollian symmetry, JHEP05 (2019) 108 [arXiv:1901.10147] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat holography: aspects of the dual field theory, JHEP12 (2016) 147 [arXiv:1609.06203] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.BITS-PilarliKK Birla Goa CampusZuarinagarIndia

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