Skip to main content
Download PDF

An action for extended string Newton-Cartan gravity

Journal of High Energy Physics volume 2019, Article number: 178 (2019) Cite this article

A preprint version of the article is available at arXiv.

Abstract

We construct an action for four-dimensional extended string Newton-Cartan gravity which is an extension of the string Newton-Cartan gravity that underlies nonrelativistic string theory. The action can be obtained as a nonrelativistic limit of the EinsteinHilbert action in General Relativity augmented with a term that contains an auxiliary two-form and one-form gauge field that both have zero flux on-shell. The four-dimensional extended string Newton-Cartan gravity is based on a central extension of the algebra that underlies string Newton-Cartan gravity.

The construction is similar to the earlier construction of a three-dimensional Chern-Simons action for extended Newton-Cartan gravity, which is based on a central extension of the algebra that underlies Newton-Cartan gravity. We show that this three-dimensional action is naturally obtained from the four-dimensional action by a reduction over the spatial isometry direction longitudinal to the string followed by a truncation of the extended string Newton-Cartan gravity fields. Our construction can be seen as a special case of the construction of an action for extended p-brane Newton-Cartan gravity in p + 3 dimensions.

References

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  2. E. Bergshoeff, J. Gomis and Z. Yan, Nonrelativistic String Theory and T-duality, JHEP 11 (2018) 133 [arXiv:1806.06071] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  3. C. Batlle, J. Gomis and D. Not, Extended Galilean symmetries of non-relativistic strings, JHEP 02 (2017) 049 [arXiv:1611.00026] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. J. Gomis and P.K. Townsend, The Galilean Superstring, JHEP 02 (2017) 105 [arXiv:1612.02759] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. C. Batlle, J. Gomis, L. Mezincescu and P.K. Townsend, Tachyons in the Galilean limit, JHEP 04 (2017) 120 [arXiv:1702.04792] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  6. T. Harmark, J. Hartong and N.A. Obers, Nonrelativistic strings and limits of the AdS/CFT correspondence, Phys. Rev. D 96 (2017) 086019 [arXiv:1705.03535] [INSPIRE].

  7. J. Klusoň, Remark About Non-Relativistic String in Newton-Cartan Background and Null Reduction, JHEP 05 (2018) 041 [arXiv:1803.07336] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. 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, JHEP 11 (2018) 190 [arXiv:1810.05560] [INSPIRE].

    ADS  Article  Google Scholar 

  9. A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  10. E.A. Bergshoeff, J. Gomis, J. Rosseel, C. Şimşek and Z. Yan, work in progress.

  11. G. Papageorgiou and B.J. Schroers, A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions, JHEP 11 (2009) 009 [arXiv:0907.2880] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

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

    ADS  Article  Google Scholar 

  13. 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].

  14. E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, Carroll versus Galilei Gravity, JHEP 03 (2017) 165 [arXiv:1701.06156] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. 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].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. D. Hansen, J. Hartong and N.A. Obers, An Action Principle for Newtonian Gravity, arXiv:1807.04765 [INSPIRE].

  17. 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].

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

  19. J.-M. Levy-Léblond, Galilei Group and Galilean Invariance, on Group Theory and its Applications, E.M. Loebl ed., Academic Press (2014) [DOI:https://doi.org/10.1016/B978-0-12-455152-7.50011-2].

  20. C. Duval and P.A. Horvathy, The ‘Peierls substitution’ and the exotic Galilei group, Phys. Lett. B 479 (2000) 284 [hep-th/0002233] [INSPIRE].

  21. R. Jackiw and V.P. Nair, Anyon spin and the exotic central extension of the planar Galilei group, Phys. Lett. B 480 (2000) 237 [hep-th/0003130] [INSPIRE].

  22. M. Bañados, R. Troncoso and J. Zanelli, Higher dimensional Chern-Simons supergravity, Phys. Rev. D 54 (1996) 2605 [gr-qc/9601003] [INSPIRE].

  23. J.F. Plebanski, On the separation of Einsteinian substructures, J. Math. Phys. 18 (1977) 2511 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  24. S. Gielen and D. Oriti, Classical general relativity as BF-Plebanski theory with linear constraints, Class. Quant. Grav. 27 (2010) 185017 [arXiv:1004.5371] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

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

  26. L. Mezincescu and P.K. Townsend, Anyons from Strings, Phys. Rev. Lett. 105 (2010) 191601 [arXiv:1008.2334] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. A. Gußmann, D. Sarkar and N. Wintergerst, Bulk-boundary correspondence between charged, anyonic strings and vortices, JHEP 12 (2018) 093 [arXiv:1809.06871] [INSPIRE].

    ADS  Article  Google Scholar 

  28. J. Brugues, J. Gomis and K. Kamimura, Newton-Hooke algebras, non-relativistic branes and generalized pp-wave metrics, Phys. Rev. D 73 (2006) 085011 [hep-th/0603023] [INSPIRE].

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

Affiliations

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

    Eric A. Bergshoeff & Ceyda Şimşek

  2. Institut für Theoretische Physik und Astrophysik, Julius-Maximilians-Universität Würzburg, Am Hubland, 97074, Würzburg, Germany

    Kevin T. Grosvenor

  3. Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, ON, N2L 6B9, Canada

    Ziqi Yan

Authors
  1. Eric A. Bergshoeff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kevin T. Grosvenor
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ceyda Şimşek
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ziqi Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eric A. Bergshoeff.

Additional information

ArXiv ePrint: 1810.09387

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

Verify currency and authenticity via CrossMark

Cite this article

Bergshoeff, E.A., Grosvenor, K.T., Şimşek, C. et al. An action for extended string Newton-Cartan gravity. J. High Energ. Phys. 2019, 178 (2019). https://doi.org/10.1007/JHEP01(2019)178

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP01(2019)178

Keywords

  • Bosonic Strings
  • Classical Theories of Gravity