Skip to main content
SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Journal of High Energy Physics
  3. Article

Lie algebra expansions and actions for non-relativistic gravity

  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 08 August 2019
  • volume 2019, Article number: 48 (2019)
Download PDF

You have full access to this open access article

Journal of High Energy Physics Aims and scope Submit manuscript
Lie algebra expansions and actions for non-relativistic gravity
Download PDF
  • Eric Bergshoeff1,
  • José Manuel Izquierdo2,
  • Tomás Ortín3 &
  • …
  • Luca Romano3 
  • 324 Accesses

  • 41 Citations

  • 3 Altmetric

  • Explore all metrics

  • Cite this article

A preprint version of the article is available at arXiv.

Abstract

We show that the general method of Lie algebra expansions can be applied to re-construct several algebras and related actions for non-relativistic gravity that have occurred in the recent literature. We explain the method and illustrate its applications by giving several explicit examples. The method can be generalized to include ultra-relativistic gravity and non-relativistic supergravity as well.

Download to read the full article text

Use our pre-submission checklist

Avoid common mistakes on your manuscript.

References

  1. A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP07 (2009) 037 [arXiv:0902.1385] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  2. M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, Torsional Newton-Cartan Geometry and Lifshitz Holography, Phys. Rev.D 89 (2014) 061901 [arXiv:1311.4794] [INSPIRE].

    ADS  MATH  Google Scholar 

  3. D.T. Son, Newton-Cartan Geometry and the Quantum Hall Effect, arXiv:1306.0638 [INSPIRE].

  4. 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].

    ADS  MathSciNet  Google Scholar 

  5. E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, Carroll versus Galilei Gravity, JHEP03 (2017) 165 [arXiv:1701.06156] [INSPIRE].

  6. G. Papageorgiou and B.J. Schroers, A Chern-Simons approach to Galilean quantum gravity in 2 + 1 dimensions, JHEP11 (2009) 009 [arXiv:0907.2880] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  7. E.A. Bergshoeff and J. Rosseel, Three-Dimensional Extended Bargmann Supergravity, Phys. Rev. Lett.116 (2016) 251601 [arXiv:1604.08042] [INSPIRE].

    Article  ADS  Google Scholar 

  8. J. Hartong, Y. Lei and N.A. Obers, Nonrelativistic Chern-Simons theories and three-dimensional Hořava-Lifshitz gravity, Phys. Rev.D 94 (2016) 065027 [arXiv:1604.08054] [INSPIRE].

    ADS  Google Scholar 

  9. D. Hansen, J. Hartong and N.A. Obers, Action Principle for Newtonian Gravity, Phys. Rev. Lett.122 (2019) 061106 [arXiv:1807.04765] [INSPIRE].

    Article  ADS  Google Scholar 

  10. D. Hansen, J. Hartong and N.A. Obers, Gravity between Newton and Einstein, arXiv:1904.05706 [INSPIRE].

  11. R. Andringa, E. Bergshoeff, J. Gomis and M. de Roo, ‘Stringy’ Newton-Cartan Gravity, Class. Quant. Grav.29 (2012) 235020 [arXiv:1206.5176] [INSPIRE].

  12. J. Brugues, T. Curtright, J. Gomis and L. Mezincescu, Non-relativistic strings and branes as non-linear realizations of Galilei groups, Phys. Lett.B 594 (2004) 227 [hep-th/0404175] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  13. E. Bergshoeff, J. Gomis and Z. Yan, Nonrelativistic String Theory and T-duality, JHEP11 (2018) 133 [arXiv:1806.06071] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  14. J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys.42 (2001) 3127 [hep-th/0009181] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  15. B. Julia and H. Nicolai, Null Killing vector dimensional reduction and Galilean geometrodynamics, Nucl. Phys.B 439 (1995) 291 [hep-th/9412002] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  16. E.A. Bergshoeff, K.T. Grosvenor, C. Simsek and Z. Yan, An Action for Extended String Newton-Cartan Gravity, JHEP01 (2019) 178 [arXiv:1810.09387] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  17. N. Ozdemir, M. Ozkan, O. Tunca and U. Zorba, Three-Dimensional Extended Newtonian (Super) Gravity, JHEP05 (2019) 130 [arXiv:1903.09377] [INSPIRE].

  18. M. Hatsuda and M. Sakaguchi, Wess-Zumino term for the AdS superstring and generalized Inönü-Wigner contraction, Prog. Theor. Phys.109 (2003) 853 [hep-th/0106114] [INSPIRE].

    Article  ADS  Google Scholar 

  19. J.A. de Azcárraga, J.M. Izquierdo, M. Picón and O. Varela, Generating Lie and gauge free differential (super) algebras by expanding Maurer-Cartan forms and Chern-Simons supergravity, Nucl. Phys.B 662 (2003) 185 [hep-th/0212347] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  20. J.A. de Azcárraga, J.M. Izquierdo, M. Picón and O. Varela, Expansions of algebras and superalgebras and some applications, Int. J. Theor. Phys.46 (2007) 2738 [hep-th/0703017] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  21. F. Izaurieta, E. Rodriguez and P. Salgado, Expanding Lie (super) algebras through Abelian semigroups, J. Math. Phys.47 (2006) 123512 [hep-th/0606215] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  22. J.A. de Azcárraga and J.M. Izquierdo, (p, q) D = 3 Poincaré supergravities from Lie algebra expansions, Nucl. Phys. B 854 (2012) 276 [arXiv:1107.2569] [INSPIRE].

  23. J.A. de Azcárraga, J.M. Izquierdo, J. Lukierski and M. Woronowicz, Generalizations of Maxwell (super) algebras by the expansion method, Nucl. Phys. B 869 (2013) 303 [arXiv:1210.1117] [INSPIRE].

  24. P.K. Concha and E.K. Rodríguez, Maxwell Superalgebras and Abelian Semigroup Expansion, Nucl. Phys.B 886 (2014) 1128 [arXiv:1405.1334] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  25. L. Avilés, E. Frodden, J. Gomis, D. Hidalgo and J. Zanelli, Non-Relativistic Maxwell Chern-Simons Gravity, JHEP05 (2018) 047 [arXiv:1802.08453] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  26. J. Gomis and A. Kleinschmidt, On free Lie algebras and particles in electro-magnetic fields, JHEP07 (2017) 085 [arXiv:1705.05854] [INSPIRE].

  27. J.M. Izquierdo, Lie algebra expansions and three-dimensional Galilean supergravity, talk given at the Spanish-Portuguese Relativity Meeting 2018 (EREP’18), Palencia, Spain, 4 September 2018.

  28. J.A. de Azcárraga, D. Gútiez and J.M. Izquierdo, Extended D = 3 Bargmann supergravity from a Lie algebra expansion, arXiv:1904.12786 [INSPIRE].

  29. T. Harmark, J. Hartong, L. Menculini, N.A. Obers and Z. Yan, Strings with Non-Relativistic Conformal Symmetry and Limits of the AdS/CFT Correspondence, JHEP11 (2018) 190 [arXiv:1810.05560] [INSPIRE].

  30. E.A. Bergshoeff, J. Rosseel, C. Şimsek and Z. Yan, Spacetime Geometry and Nonrelativistic String Theory, in preparation.

  31. D. Van den Bleeken, Torsional Newton-Cartan gravity from the large c expansion of general relativity, Class. Quant. Grav.34 (2017) 185004 [arXiv:1703.03459] [INSPIRE].

  32. D. Van den Bleeken, Torsional Newton-Cartan gravity and strong gravitational fields, in proceedings of the 15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics and Relativistic Field Theories (MG15), Rome, Italy, 1-7 July 2018, arXiv:1903.10682 [INSPIRE].

  33. P.D. Álvarez, J. Gomis, K. Kamimura and M.S. Plyushchay, (2 + 1) D Exotic Newton-Hooke Symmetry, Duality and Projective Phase, Annals Phys.322 (2007) 1556 [hep-th/0702014] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  34. O. Khasanov and S. Kuperstein, (In) finite extensions of algebras from their Inonu-Wigner contractions, J. Phys.A 44 (2011) 475202 [arXiv:1103.3447] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  35. M. Henneaux, Geometry of Zero Signature Space-times, Bull. Soc. Math. Belg.31 (1979) 47 [INSPIRE].

    MathSciNet  MATH  Google Scholar 

  36. E. Bergshoeff, D. Grumiller, S. Prohazka and J. Rosseel, Three-dimensional Spin-3 Theories Based on General Kinematical Algebras, JHEP01 (2017) 114 [arXiv:1612.02277] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

Author information

Authors and Affiliations

  1. Van Swinderen Institute, University of Groningen, Nijenborgh 4, Groningen, 9747, AG, The Netherlands

    Eric Bergshoeff

  2. Departamento de Fisica Teorica, Universidad de Valladolid, Paseo de Belén 11, 46011, Valladolid, Spain

    José Manuel Izquierdo

  3. Instituto de Física Teórica UAM/CSIC, C/ Nicolás Cabrera, 13-15, C.U. Cantoblanco, E-28049, Madrid, Spain

    Tomás Ortín & Luca Romano

Authors
  1. Eric Bergshoeff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José Manuel Izquierdo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tomás Ortín
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Luca Romano
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric Bergshoeff.

Additional information

ArXiv ePrint: 1904.08304

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bergshoeff, E., Izquierdo, J.M., Ortín, T. et al. Lie algebra expansions and actions for non-relativistic gravity. J. High Energ. Phys. 2019, 48 (2019). https://doi.org/10.1007/JHEP08(2019)048

Download citation

  • Received: 19 May 2019

  • Revised: 10 July 2019

  • Accepted: 24 July 2019

  • Published: 08 August 2019

  • DOI: https://doi.org/10.1007/JHEP08(2019)048

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Classical Theories of Gravity
  • Space-Time Symmetries
  • Chern-Simons Theories
  • Effective Field Theories
Use our pre-submission checklist

Avoid common mistakes on your manuscript.

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

Not affiliated

Springer Nature

© 2023 Springer Nature